ThanhTam
23-10-2009, 07:47 AM
Các BDT liên quan đến đường phân giác
CÁC BẤT ĐẲNG THỨC LIÊN QUAN ĐẾN PHAN GIÁC
Cho [Only registered and activated users can see links] Gọi Ao là tâm đường tròn bàng tiếp góc A, ; A1 và A2 lần lượt giao điểm của phân giác trong góc A với đường tròn ngoại tiếp tam giác ABC và cạnh BC .Các điểm Bi , Ci (i=0,1,2) được định nghĩa tương tự .Gọi I là tâm đường tròn nội tiếp tam giác ABC. Khi đó ta có các tính chất sau:
Tính chất 1: [Only registered and activated users can see links] %5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7Bb%20+%20c%7D%7D; [Only registered and activated users can see links] %5Cfrac%7BB%7D%7B2%7D%7D%7D%7B%7Bc%20+%20a%7D%7D ; [Only registered and activated users can see links] %5Cfrac%7BC%7D%7B2%7D%7D%7D%7B%7Ba%20+%20b%7D%7D
Tính chất 2: [Only registered and activated users can see links] 7D%7B%7B%5Ccos%20%5Cfrac%7BA%7D%7B2%7D%7D%7D ; [Only registered and activated users can see links] 7D%7B%7B%5Ccos%20%5Cfrac%7BB%7D%7B2%7D%7D%7D ; [Only registered and activated users can see links] 7D%7B%7B%5Ccos%20%5Cfrac%7BC%7D%7B2%7D%7D%7D
Tính chất 3: [Only registered and activated users can see links] 7B2%7D%5Ccos%20%5Cfrac%7BC%7D%7B2%7D ; [Only registered and activated users can see links] 7B2%7D%5Ccos%20%5Cfrac%7BA%7D%7B2%7D ; [Only registered and activated users can see links] 7B2%7D%5Ccos%20%5Cfrac%7BB%7D%7B2%7D
Tính chất 4: [Only registered and activated users can see links] D%5Csin%20%5Cfrac%7BC%7D%7B2%7D ; [Only registered and activated users can see links] D%5Csin%20%5Cfrac%7BA%7D%7B2%7D ; [Only registered and activated users can see links] D%5Csin%20%5Cfrac%7BB%7D%7B2%7D
[Only registered and activated users can see links]
Việc chứng minh các tính chất này xin dành cho bạn đọc
Bây giờ ta xét một số bài toán sau:
Bài toán 1: Ñaët [Only registered and activated users can see links] %7B%5Crm%7B%20=%20%7D%7D%5Cfrac%7B%7B%7B%5Crm%7BAA %7D%7D_%7B%5Crm%7B2%7D%7D%7D%7D%7B%7B%7B%5Crm%7BAA %7D%7D_%7B%5Crm%7B1%7D%7D%20%7D%7D ; [Only registered and activated users can see links] %7B%5Crm%7B%20=%20%7D%7D%5Cfrac%7B%7B%7B%5Crm%7BBB %7D%7D_%7B%5Crm%7B2%7D%7D%7D%7D%7B%7B%7B%5Crm%7BBB %7D%7D_%7B%5Crm%7B1%7D%7D%20%7D%7D ; [Only registered and activated users can see links] %7B%5Crm%7B%20=%20%7D%7D%5Cfrac%7B%7B%7B%5Crm%7BCC %7D%7D_%7B%5Crm%7B2%7D%7D%7D%7D%7B%7B%7B%5Crm%7BCC %7D%7D_%7B%5Crm%7B1%7D%7D%20%7D%7D
CMR: a) [Only registered and activated users can see links] %20%5Cfrac%7B9%7D%7B4%7D ;
b) [Only registered and activated users can see links] 5Cfrac%7B1%7D%7B%7Bl_b%20%7D%7D%20+%20%5Cfrac%7B1% 7D%7B%7Bl_c%20%7D%7D%20%5Cge%204
c/[Only registered and activated users can see links] %5E2%20A%7D%7D%20+%20%5Cfrac%7B%7Bl_b%20%7D%7D%7B% 7B%5Csin%20%5E2%20B%7D%7D%20+%20%5Cfrac%7B%7Bl_c%7 D%7D%7B%7B%5Csin%20%5E2%20C%7D%7D%20%5Cge%203 (Olympic chaâu AÙ 1998)
Lời giaûi:
a)Ta có : [Only registered and activated users can see links] 5CRightarrow [Only registered and activated users can see links] %7B%5Crm%7B%20=%20%7D%7D%5Cfrac%7B%7B%7B%5Crm%7B4b c%7D%7D%7B%5Crm%7B.cos%7D%7D%5E%7B%5Crm%7B2%7D%7D% 5Cfrac%7B%7B%5Crm%7BA%7D%7D%7D%7B%7B%5Crm%7B2%7D%7 D%7D%7D%7D%7B%7B%7B%5Crm%7B%28b%20+%20c%29%7D%7D%5 E%7B%5Crm%7B2%7D%7D%20%7D%7D%20%5Cle%7B%5Crm%7Bcos %7D%7D%5E%7B%5Crm%7B2%7D%7D%20%5Cfrac%7B%7B%5Crm%7 BA%7D%7D%7D%7B%7B%5Crm%7B2%7D%7D%7D chứng minh tương tự ta có: [Only registered and activated users can see links] %20%5Cle%20%7B%5Crm%7Bcos%7D%7D%5E%7B%5Crm%7B2%7D% 7D%5Cfrac%7B%7B%5Crm%7BB%7D%7D%7D%7B%7B%5Crm%7B2%7 D%7D%7D;%7B%5Crm%7B%20l%7D%7D_%7B%5Crm%7Bb%7D%7D%2 0%20%5Cle%20%7B%5Crm%7Bcos%7D%7D%5E%7B%5Crm%7B2%7D %7D%5Cfrac%7B%7B%5Crm%7BC%7D%7D%7D%7B%7B%5Crm%7B2% 7D%7D%7D
[Only registered and activated users can see links] 0+%20l_c%20%20%5Cle%20%5Ccos%20%5E2%20%5Cfrac%7BA% 7D%7B2%7D%20+%20%5Ccos%20%5E2%5Cfrac%7BB%7D%7B2%7D %20+%20%5Ccos%20%5E2%20%5Cfrac%7BC%7D%7B2%7D%20%5C le%20%5Cfrac%7B9%7D%7B4%7D
b/ Ta có : [Only registered and activated users can see links] %20%20+%20%7B%5Crm%7Bl%7D%7D_%7B%5Crm%7Bb%7D%7D%20 %20+%20%7B%5Crm%7Bl%7D%7D_%7B%5Crm%7Bc%7D%7D%7B%5C rm%7B%29%28%7D%7D%5Cfrac%7B%7B%5Crm%7B1%7D%7D%7D%7 B%7B%7B%5Crm%7Bl%7D%7D_%7B%5Crm%7Ba%7D%7D%20%7D%7D %20+%5Cfrac%7B%7B%5Crm%7B1%7D%7D%7D%7B%7B%7B%5Crm% 7Bl%7D%7D_%7B%5Crm%7Bb%7D%7D%20%7D%7D%20+%5Cfrac%7 B%7B%5Crm%7B1%7D%7D%7D%7B%7B%7B%5Crm%7Bl%7D%7D_%7B %5Crm%7Bc%7D%7D%20%7D%7D%7B%5Crm%7B%29%7D%7D%20%5C ge%209
[Only registered and activated users can see links] _a%20%7D%7D%20+%20%5Cfrac%7B1%7D%7B%7Bl_b%20%7D%7D %20+%20%5Cfrac%7B1%7D%7B%7Bl_c%20%7D%7D%5Cge%204
c)Ta có:
[Only registered and activated users can see links] 5E2%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B%28b%20+%20 c%29%5E2%20%7D%7D%20=%5Cfrac%7B%7B4.2R%5Csin%20B.2 R%5Csin%20C.%5Ccos%20%5E2%20%5Cfrac%7BA%7D%7B2%7D% 7D%7D%7B%7B%282R%5Csin%20B%20+%202R%5CsinC%29%5E2% 20%7D%7D%20=%20%5Cfrac%7B%7B4%5Csin%20B.%5Csin%20C .%5Ccos%20%5E2%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B %28%5Csin%20B%20+%20%5CsinC%29%5E2%20%7D%7D%20=%20 %5Cfrac%7B%7B4%5Csin%20B.%5Csin%20C.%5Ccos%20%5E2% 20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B4%5Ccos%20%5E2% 5Cfrac%7BA%7D%7B2%7D.%5Ccos%20%5E2%20%5Cfrac%7B%7B B%20-%20C%7D%7D%7B2%7D%7D%7D
[Only registered and activated users can see links] 7B%7B%5Csin%20B.%5Csin%20C%7D%7D%7B%7B%5Ccos%20%5E 2%20%5Cfrac%7B%7BB%20-C%7D%7D%7B2%7D%7D%7D%20%5Cge%20%5Csin%20B.%5Csin%2 0C%20%5CRightarrow%20%5Cfrac%7B%7Bl_a%20%7D%7D%7B% 7B%5Csin%20%5E2%20A%7D%7D%20%5Cge%5Cfrac%7B%7B%5Cs in%20B.%5Csin%20C%7D%7D%7B%7B%5Csin%20%5E2%20A%7D% 7D .
Chứng minh tương tự ta cũng có
[Only registered and activated users can see links] %5E2%20B%7D%7D%20%5Cge%20%5Cfrac%7B%7B%5Csin%20C.% 5Csin%20A%7D%7D%7B%7B%5Csin%20%5E2B%7D%7D%7B%5Crm% 7B%20;%20%20%20%20%7D%7D%5Cfrac%7B%7B%7B%5Crm%7Bl% 7D%7D_%7B%5Crm%7Bc%7D%7D%20%7D%7D%7B%7B%7B%5Crm%7B sin%7D%7D%5E%7B%5Crm%7B2%7D%7D%20C%7D%7D%5Cge%20%5 Cfrac%7B%7B%5Csin%20A.%5Csin%20B%7D%7D%7B%7B%5Csin %20%5E2%20C%7D%7D
[Only registered and activated users can see links] D%7D%7B%7B%5Csin%20%5E2%20A%7D%7D%20+%20%5Cfrac%7B %7Bl_b%20%7D%7D%7B%7B%5Csin%20%5E2%20B%7D%7D+%20%5 Cfrac%7B%7Bl_c%20%7D%7D%7B%7B%5Csin%20%5E2%20C%7D% 7D%20%5Cge%20%5Cfrac%7B%7B%5Csin%20B.%5Csin%20C%7D %7D%7B%7B%5Csin%20%5E2%20A%7D%7D%20+%5Cfrac%7B%7B% 5Csin%20C.%5Csin%20A%7D%7D%7B%7B%5Csin%20%5E2%20B% 7D%7D%20+%20%5Cfrac%7B%7B%5Csin%20A.%5Csin%20B%7D% 7D%7B%7B%5Csin%20%5E2C%7D%7D%20%5Cge%203%5Csqrt[3]%7B%7B%5Cfrac%7B%7B%5Csin%20%5E2%20%7B%5Crm%7BA%7D %7D%7B%5Crm%7B.sin%7D%7D%5E%7B%5Crm%7B2%7D%7D%20B. %5Csin%5E2%20C%7D%7D%7B%7B%5Csin%20%5E2%20%7B%5Crm %7BA%7D%7D%7B%5Crm%7B.sin%7D%7D%5E%7B%5Crm%7B2%7D% 7D%20B.%5Csin%20%5E2%20C%7D%7D%7D%7D%20=%203
Đẳng thức xảy ra khi và chỉ khi A = B = C
Bài toán 2 : Đặt [Only registered and activated users can see links] 7B%7BAA_0%20%7D%7D;%7B%5Crm%7B%20%7D%7Dx_b%20%20=% 20%5Cfrac%7B%7BBB_2%20%7D%7D%7B%7BBB_0%7D%7D;%7B%5 Crm%7B%20%7D%7Dx_c%20%20=%20%5Cfrac%7B%7BCC_2%20%7 D%7D%7B%7BCC_0%20%7D%7D. Cmr:
a) [Only registered and activated users can see links] 20+%20x_c%20%20%5Cle%20%5Cfrac%7B3%7D%7B2%7D
[Only registered and activated users can see links] rm%7B1%7D%7D%7D%7B%7B%7B%5Crm%7Bx%7D%7D_%7B%5Crm%7 Ba%7D%7D%20%7D%7D%20+%20%5Cfrac%7B1%7D%7B%7Bx_b%20 %7D%7D%20+%5Cfrac%7B1%7D%7B%7Bx_c%20%7D%7D%20%5Cge %206%7B%5Crm%7B%20%7D%7D
[Only registered and activated users can see links] %7B%5Crm%7Bx%7D%7D_%7B%5Crm%7Ba%7D%7D%20%7D%7D%7B% 7B%5Csin%20%5E2%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20+% 5Cfrac%7B%7B%7B%5Crm%7Bx%7D%7D_%7B%5Crm%7Bb%7D%7D% 20%7D%7D%7B%7B%5Csin%20%5E2%20%5Cfrac%7BB%7D%7B2%7 D%7D%7D%20+%5Cfrac%7B%7B%7B%5Crm%7Bx%7D%7D_%7B%5Cr m%7Bc%7D%7D%20%7D%7D%7B%7B%5Csin%20%5E2%20%5Cfrac% 7BC%7D%7B2%7D%7D%7D%20%5Cge%206
Lời giải:
a) Ta có: [Only registered and activated users can see links] 7B%7BAA_0%20%7D%7D%20=%20%5Cfrac%7B%7B2bc.%5Ccos%2 0%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B4R%28b+%20c%29.% 5Ccos%20%5Cfrac%7BB%7D%7B2%7D.%5Ccos%20%5Cfrac%7BC %7D%7B2%7D%7D%7D%20=%20%5Cfrac%7B%7B%5Csin%20B.%5C sin%20C.%5Ccos%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B%28 %5Csin%20B%20+%20%5Csin%20C%29.%5Ccos%20%5Cfrac%7B B%7D%7B2%7D.%5Ccos%20%5Cfrac%7BC%7D%7B2%7D%7D%7D%2 0=
[Only registered and activated users can see links] B%7D%7B2%7D.%5Csin%20%5Cfrac%7BC%7D%7B2%7D.%5Ccos% 20%5Cfrac%7BB%7D%7B2%7D.%5Ccos%5Cfrac%7BC%7D%7B2%7 D.%5Ccos%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B2%5Cco s%20%5Cfrac%7BA%7D%7B2%7D.%5Ccos%20%5Cfrac%7B%7BB% 20-C%7D%7D%7B2%7D.%5Ccos%20%5Cfrac%7BB%7D%7B2%7D.%5Cc os%20%5Cfrac%7BC%7D%7B2%7D%7D%7D%20=%20%5Cfrac%7B% 7B2%5Csin%5Cfrac%7BB%7D%7B2%7D.%5Csin%20%5Cfrac%7B C%7D%7B2%7D%7D%7D%7B%7B%5Ccos%20%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D
[Only registered and activated users can see links] 7BB%20-%20C%7D%7D%7B2%7D%20-%20%5Csin%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B%5Cco s%20%5Cfrac%7B%7BB-%20C%7D%7D%7B2%7D%7D%7D%20=%201%20-%20%5Cfrac%7B%7B%5Csin%20%5Cfrac%7BA%7D%7B2%7D%7D% 7D%7B%7B%5Ccos%20%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D
Tương tự: [Only registered and activated users can see links] 7D%7B%7B%5Ccos%20%5Cfrac%7B%7BC%20-%20A%7D%7D%7B2%7D%7D%7D%7B%5Crm%7B%20%7D%7D;%7B%5C rm%7B%20%20%20%20x%7D%7D_%7B%5Crm%7Bc%7D%7D%20%7B% 5Crm%7B%20=%20%7D%7D1%20-%5Cfrac%7B%7B%7B%5Crm%7Bsin%7D%7D%5Cfrac%7B%7B%5Cr m%7BC%7D%7D%7D%7B%7B%5Crm%7B2%7D%7D%7D%7D%7D%7B%7B %5Ccos%20%5Cfrac%7B%7BA%20-B%7D%7D%7B2%7D%7D%7D%7B%5Crm%7B%20%7D%7D
[Only registered and activated users can see links] 0+%20x_c%20%20=%203%20-%20%28%5Cfrac%7B%7B%5Csin%5Cfrac%7BA%7D%7B2%7D%7D% 7D%7B%7B%5Ccos%20%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B%7B%5Csin% 5Cfrac%7BB%7D%7B2%7D%7D%7D%7B%7B%5Ccos%20%5Cfrac%7 B%7BC%20-%20A%7D%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B%7B%5Csin% 5Cfrac%7BC%7D%7B2%7D%7D%7D%7B%7B%5Ccos%20%5Cfrac%7 B%7BA%20-%20B%7D%7D%7B2%7D%7D%7D%29
Mặt khác: [Only registered and activated users can see links] %7D%7D%7D%7B%7B%5Ccos%20%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B%7B%5Csin% 5Cfrac%7BB%7D%7B2%7D%7D%7D%7B%7B%5Ccos%20%5Cfrac%7 B%7BC%20-%20A%7D%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B%7B%5Csin% 5Cfrac%7BC%7D%7B2%7D%7D%7D%7B%7B%5Ccos%20%5Cfrac%7 B%7BA%20-%20B%7D%7D%7B2%7D%7D%7D
[Only registered and activated users can see links] 7B%5Csin%20B%20+%20%5Csin%20C%7D%7D%20+%20%5Cfrac% 7B%7B%5Csin%20B%7D%7D%7B%7B%5Csin%20C%20+%5Csin%20 A%7D%7D%20+%20%5Cfrac%7B%7B%5Csin%20C%7D%7D%7B%7B% 5Csin%20A%20+%20%5Csin%20B%7D%7D%20%5Cge%20%5Cfrac %7B3%7D%7B2%7D
[Only registered and activated users can see links] 0+%20x_c%20%20%5Cle%203%20-%20%5Cfrac%7B3%7D%7B2%7D%20=%20%5Cfrac%7B3%7D%7B2% 7D. Đẳng thức xảy ra khi và chỉ khi tam giác ABC đều
b)Từ Bđt : [Only registered and activated users can see links] %20%20+%20x_%7B%5Crm%7Bb%7D%7D%20%20+%20x_%7B%5Crm %7Bc%7D%7D%7B%5Crm%7B%29%28%7D%7D%5Cfrac%7B%7B%5Cr m%7B1%7D%7D%7D%7B%7Bx_%7B%5Crm%7Ba%7D%7D%20%7D%7D% 20+%20%5Cfrac%7B%7B%5Crm%7B1%7D%7D%7D%7B%7Bx_%7B%5 Crm%7Bb%7D%7D%7D%7D%20+%20%5Cfrac%7B%7B%5Crm%7B1%7 D%7D%7D%7B%7Bx_%7B%5Crm%7Bc%7D%7D%20%7D%7D%7B%5Crm %7B%29%7D%7D%20%5Cge%209%20%5CRightarrow%5Cfrac%7B %7B%5Crm%7B1%7D%7D%7D%7B%7Bx_%7B%5Crm%7Ba%7D%7D%20 %7D%7D%20+%20%5Cfrac%7B%7B%5Crm%7B1%7D%7D%7D%7B%7B x_%7B%5Crm%7Bb%7D%7D%20%7D%7D%20+%5Cfrac%7B%7B%5Cr m%7B1%7D%7D%7D%7B%7Bx_%7B%5Crm%7Bc%7D%7D%20%7D%7D% 20%5Cge%20%5Cfrac%7B9%7D%7B%7B%7B%5Crm%7Bx%7D%7D_% 7B%5Crm%7Ba%7D%7D%20%20+x_%7B%5Crm%7Bb%7D%7D%20%20 +%20x_%7B%5Crm%7Bc%7D%7D%20%7D%7D%20%5Cge%206 (đpcm)
c)Ta có: [Only registered and activated users can see links] rac%7BB%7D%7B2%7D.%5Csin%20%5Cfrac%7BC%7D%7B2%7D%7 D%7D%7B%7B%5Ccos%20%5Cfrac%7B%7BB%20-C%7D%7D%7B2%7D%7D%7D%20%5Cge%202%5Csin%20%5Cfrac%7 BB%7D%7B2%7D.%5Csin%20%5Cfrac%7BC%7D%7B2%7D%20%5CR ightarrow%20%5Cfrac%7B%7Bx_a%7D%7D%7B%7B%5Csin%20% 5E2%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20%5Cge%202%5Cfr ac%7B%7B%5Csin%20%5Cfrac%7BB%7D%7B2%7D.%5Csin%5Cfr ac%7BC%7D%7B2%7D%7D%7D%7B%7B%5Csin%20%5E2%20%5Cfra c%7BA%7D%7B2%7D%7D%7D.
Chứng minh tương tự ta có:
[Only registered and activated users can see links] %5E2%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20%5Cge%202%5Cf rac%7B%7B%5Csin%5Cfrac%7BC%7D%7B2%7D.%5Csin%20%5Cf rac%7BA%7D%7B2%7D%7D%7D%7B%7B%5Csin%20%5E2%20%5Cfr ac%7BB%7D%7B2%7D%7D%7D%7B%5Crm%7B%20%20;%20%7D%7D% 5Cfrac%7B%7Bx_c%20%7D%7D%7B%7B%5Csin%20%5E2%20%5Cf rac%7BC%7D%7B2%7D%7D%7D%20%5Cge%202%5Cfrac%7B%7B%5 Csin%5Cfrac%7BA%7D%7B2%7D.%5Csin%20%5Cfrac%7BB%7D% 7B2%7D%7D%7D%7B%7B%5Csin%20%5E2%20%5Cfrac%7BC%7D%7 B2%7D%7D%7D
[Only registered and activated users can see links] D%7D%7B%7B%5Csin%20%5E2%20%5Cfrac%7BA%7D%7B2%7D%7D %7D%20+%20%5Cfrac%7B%7Bx_b%7D%7D%7B%7B%5Csin%20%5E 2%20%5Cfrac%7BB%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B%7 Bx_c%20%7D%7D%7B%7B%5Csin%20%5E2%20%5Cfrac%7BC%7D% 7B2%7D%7D%7D
[Only registered and activated users can see links] Csin%20%5Cfrac%7BB%7D%7B2%7D.%5Csin%20%5Cfrac%7BC% 7D%7B2%7D%7D%7D%7B%7B%5Csin%20%5E2%5Cfrac%7BA%7D%7 B2%7D%7D%7D%20+%20%5Cfrac%7B%7B%5Csin%20%5Cfrac%7B C%7D%7B2%7D.%5Csin%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7 B%7B%5Csin%20%5E2%5Cfrac%7BB%7D%7B2%7D%7D%7D%20+%2 0%5Cfrac%7B%7B%5Csin%20%5Cfrac%7BA%7D%7B2%7D.%5Csi n%20%5Cfrac%7BB%7D%7B2%7D%7D%7D%7B%7B%5Csin%20%5E2 %5Cfrac%7BC%7D%7B2%7D%7D%7D%7D%20%5Cright%29%20%5C ge%202.3%20=%206.
Bài 3: Chứng minh rằng
[Only registered and activated users can see links] 20%20%5Cge%20IA%20+%20IB%20+%20IC
[Only registered and activated users can see links] D%7B%7BIA%7D%7D%20+%20%5Cfrac%7B1%7D%7B%7BIB%7D%7D %20+%20%5Cfrac%7B1%7D%7B%7BIC%7D%7D%20%5Cge%5Cfrac %7B1%7D%7B%7BIA_1%20%7D%7D%20+%20%5Cfrac%7B1%7D%7B %7BIB_1%20%7D%7D%20+%20%5Cfrac%7B1%7D%7B%7BIC_1%20 %7D%7D%20%5Cge%5Cfrac%7B3%7D%7BR%7D%20%5C%5C%20%20 %7B%5Crm%7B%20%20%20%7D%7D%20%5C%5C%20%20%5Cend%7B array%7D
Lời giải : a) Ta có: [Only registered and activated users can see links] cos%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20-%204R%5Csin%5Cfrac%7BB%7D%7B2%7D%5Csin%20%5Cfrac%7 BC%7D%7B2%7D%20=%20%5Cfrac%7B%7BR%28%5Csin%20B%20+ %20%5Csin%20C%29%7D%7D%7B%7B%5Ccos%5Cfrac%7BA%7D%7 B2%7D%7D%7D%20-%204R%5Csin%20%5Cfrac%7BB%7D%7B2%7D%5Csin%20%5Cfra c%7BC%7D%7B2%7D
[Only registered and activated users can see links] c%7BC%7D%7B2%7D%20=2R%5Csin%20%5Cfrac%7BA%7D%7B2%7 D%7B%5Crm%7B%20%20%20%20%7D%7DT%5C
[Only registered and activated users can see links] %20+%20IC_1%20%20=%202R%28%5Csin%20%5Cfrac%7BA%7D% 7B2%7D%20+%20%5Csin%5Cfrac%7BB%7D%7B2%7D%20+%20%5C sin%20%5Cfrac%7BC%7D%7B2%7D%29
[Only registered and activated users can see links]
[Only registered and activated users can see links] %20%20%7D%7D%20%5Cge%202%28%5Csin%20%5Cfrac%7BA%7D %7B2%7D.%5Csin%20%5Cfrac%7BA%7D%7B2%7D%20+%20%5Csi n%5Cfrac%7BB%7D%7B2%7D.%5Csin%20%5Cfrac%7BC%7D%7B2 %7D%20+%20%5Csin%20%5Cfrac%7BC%7D%7B2%7D.%5Csin%20 %5Cfrac%7BA%7D%7B2%7D%29%20%5C%5C%20%20%5Cend%7Bar ray%7D
[Only registered and activated users can see links] %20+%20IC_1%20%20%5Cge%204R%28%5Csin%20%5Cfrac%7BA %7D%7B2%7D%5Csin%5Cfrac%7BB%7D%7B2%7D%20+%20%5Csin %20%5Cfrac%7BB%7D%7B2%7D%5Csin%20%5Cfrac%7BC%7D%7B 2%7D%20+%20%5Csin%20%5Cfrac%7BC%7D%7B2%7D%5Csin%5C frac%7BA%7D%7B2%7D%29%20=%20IA%20+%20IB%20+%20IC đpcm
[Only registered and activated users can see links] 7D%7D%5Cfrac%7B1%7D%7B%7BIA%7D%7D%20+%20%5Cfrac%7B 1%7D%7B%7BIB%7D%7D%20+%5Cfrac%7B1%7D%7B%7BIC%7D%7D %20=%20%5Cfrac%7B1%7D%7B%7B4R%7D%7D%28%5Cfrac%7B1% 7D%7B%7B%5Csin%20%5Cfrac%7BA%7D%7B2%7D%5Csin%5Cfra c%7BB%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B1%7D%7B%7B%5 Csin%20%5Cfrac%7BB%7D%7B2%7D%5Csin%20%5Cfrac%7BC%7 D%7B2%7D%7D%7D%20+%5Cfrac%7B1%7D%7B%7B%5Csin%20%5C frac%7BC%7D%7B2%7D%5Csin%20%5Cfrac%7BA%7D%7B2%7D%7 D%7D%29%20=%20%20%5C%5C
[Only registered and activated users can see links] Cfrac%7B%7B%5Csin%20%5Cfrac%7BA%7D%7B2%7D%20+%20%5 Csin%20%5Cfrac%7BB%7D%7B2%7D%20+%20%5Csin%5Cfrac%7 BC%7D%7B2%7D%7D%7D%7B%7B%5Csin%20%5Cfrac%7BA%7D%7B 2%7D%5Csin%20%5Cfrac%7BB%7D%7B2%7D%5Csin%20%5Cfrac %7BC%7D%7B2%7D%7D%7D%20%5Cge%5Cfrac%7B1%7D%7B%7B4R %7D%7D.%5Cfrac%7B%7B2%28%5Csin%20%5Cfrac%7BA%7D%7B 2%7D.%5Csin%20%5Cfrac%7BA%7D%7B2%7D%20+%20%5Csin%5 Cfrac%7BB%7D%7B2%7D.%5Csin%20%5Cfrac%7BC%7D%7B2%7D %20+%20%5Csin%20%5Cfrac%7BC%7D%7B2%7D.%5Csin%5Cfra c%7BA%7D%7B2%7D%29%7D%7D%7B%7B%5Csin%20%5Cfrac%7BA %7D%7B2%7D%5Csin%20%5Cfrac%7BB%7D%7B2%7D%5Csin%20% 5Cfrac%7BC%7D%7B2%7D%7D%7D%20=
[Only registered and activated users can see links] 0%5Cfrac%7BA%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B1%7D% 7B%7B2R%5Csin%20%5Cfrac%7BB%7D%7B2%7D%7D%7D+%20%5C frac%7B1%7D%7B%7B2R%5Csin%20%5Cfrac%7BC%7D%7B2%7D% 7D%7D%20=%20%5Cfrac%7B1%7D%7B%7BIA_1%20%7D%7D%20+% 20%5Cfrac%7B1%7D%7B%7BIB_1%7D%7D%20+%20%5Cfrac%7B1 %7D%7B%7BIC_1%20%7D%7D%20%5C%5C%20%20%5Cend%7Barra y%7D
Bài toán 4:Chứng minh rằng:
[Only registered and activated users can see links] %20B_1%20B_2%20%20+%20C_1%20C_2%20%20%5Cge%20%5Cfr ac%7B%7B2p%5E2%20%7D%7D%7B%7B9R%7D%7D
[Only registered and activated users can see links] 0+%20C_2%20C_0%20%20%5Cge%20%5Csqrt%203%20p
a)Ta có : [Only registered and activated users can see links] B2%5Ccos%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20-%5Cfrac%7B%7B2bc.%5Ccos%20%5Cfrac%7BA%7D%7B2%7D%7D %7D%7B%7Bb%20+%20c%7D%7D%20=%20%5Cfrac%7B%7B%28b%2 0+%20c%29%5E2%20%20-%202bc.%5Ccos%5E2%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B %7B2%28b%20+%20c%29%5Ccos%20%5Cfrac%7BA%7D%7B2%7D% 7D%7D%20=
[Only registered and activated users can see links] 5E2%20%20-%204bc.%5Ccos%20A%7D%7D%7B%7B2%28b%20+%20c%29%5Cco s%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20=%5Cfrac%7B%7Ba% 5E2%20%7D%7D%7B%7B8R%28%5Csin%20B%20+%20%5Csin%20C %29%5Ccos%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20=%20%5Cf rac%7B%7Ba%5E2%7D%7D%7B%7B8R.%5Ccos%20%5E2%20%5Cfr ac%7BA%7D%7B2%7D.%5Ccos%20%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D%20%5Cge%20%5Cfrac%7B%7Ba%5 E2%7D%7D%7B%7B8R.%5Ccos%20%5E2%20%5Cfrac%7BA%7D%7B 2%7D%7D%7D
Tương tự[Only registered and activated users can see links] 20%5Cge%20%5Cfrac%7B%7Bb%5E2%20%7D%7D%7B%7B8R.%5Cc os%20%5E2%20%5Cfrac%7BB%7D%7B2%7D%7D%7D%7B%5Crm%7B %20%7D%7D;%7B%5Crm%7B%20%20C%7D%7D_%7B%5Crm%7B1%7D %7D%20%7B%5Crm%7BC%7D%7D_%7B%5Crm%7B2%7D%7D%20%20% 5Cge%5Cfrac%7B%7B%7B%5Crm%7Bc%7D%7D%5E%7B%5Crm%7B2 %7D%7D%20%7D%7D%7B%7B8R.%5Ccos%20%5E2%20%5Cfrac%7B C%7D%7B2%7D%7D%7D%7B%5Crm%7B%20%7D%7D
[Only registered and activated users can see links] 1%20B_2%20%20+%20C_1%20C_2%20%20%5Cge%20%5Cfrac%7B 1%7D%7B%7B8R%7D%7D%5Cleft%28%7B%5Cfrac%7B%7Ba%5E2% 20%7D%7D%7B%7B%5Ccos%20%5E2%20%5Cfrac%7BA%7D%7B2%7 D%7D%7D%20+%20%5Cfrac%7B%7Bb%5E2%20%7D%7D%7B%7B%5C cos%20%5E2%5Cfrac%7BB%7D%7B2%7D%7D%7D%20+%20%5Cfra c%7B%7Bc%5E2%20%7D%7D%7B%7B%5Ccos%20%5E2%20%5Cfrac %7BC%7D%7B2%7D%7D%7D%7D%20%5Cright%29
Mặt khác áp dụng Bđt Bunhia ta có:
[Only registered and activated users can see links] 7BA%7D%7B2%7D%20+%20%5Ccos%20%5E2%20%5Cfrac%7BB%7D %7B2%7D%20+%20%5Ccos%20%5E2%5Cfrac%7BC%7D%7B2%7D%7 D%20%5Cright%29%5Cleft%28%20%7B%5Cfrac%7B%7Ba%5E2% 20%7D%7D%7B%7B%5Ccos%20%5E2%20%5Cfrac%7BA%7D%7B2%7 D%7D%7D%20+%5Cfrac%7B%7Bb%5E2%20%7D%7D%7B%7B%5Ccos %20%5E2%20%5Cfrac%7BB%7D%7B2%7D%7D%7D%20+%20%5Cfra c%7B%7Bc%5E2%20%7D%7D%7B%7B%5Ccos%20%5E2%5Cfrac%7B C%7D%7B2%7D%7D%7D%7D%20%5Cright%29%20%5Cge%20%28a% 20+%20b%20+%20c%29%5E2
[Only registered and activated users can see links] 1%20B_2%20%20+%20C_1%20C_2%20%20%5Cge%5Cfrac%7B1%7 D%7B%7B8R%7D%7D%5Cfrac%7B%7B%28a%20+%20b%20+%20c%2 9%5E2%20%7D%7D%7B%7B%5Ccos%20%5E2%20%5Cfrac%7BA%7D %7B2%7D%20+%20%5Ccos%20%5E2%5Cfrac%7BB%7D%7B2%7D%2 0+%20%5Ccos%20%5E2%20%5Cfrac%7BC%7D%7B2%7D%7D%7D%2 0%5Cge%20%5Cfrac%7B%7B4p%5E2%7D%7D%7B%7B8R.%5Cfrac %7B9%7D%7B4%7D%7D%7D%20=%20%5Cfrac%7B%7B2p%5E2%20% 7D%7D%7B%7B9R%7D%7D đpcm
b)Ta có:
[Only registered and activated users can see links] %5Ccos%20%5Cfrac%7BC%7D%7B2%7D%20-%5Cfrac%7B%7B2bc.%5Ccos%20%5Cfrac%7BA%7D%7B2%7D%7D %7D%7B%7Bb%20+%20c%7D%7D%20=%204R%5Ccos%20%5Cfrac% 7BB%7D%7B2%7D%5Ccos%5Cfrac%7BC%7D%7B2%7D%20-%20%5Cfrac%7B%7B4R%5Csin%20B%5Csin%20C%7D%7D%7B%7B 2%5Ccos%20%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D%20=
[Only registered and activated users can see links] 5Ccos%20%5Cfrac%7BC%7D%7B2%7D%5Cleft%28%20%7B1%20-%20%5Cfrac%7B%7B2%5Csin%5Cfrac%7BB%7D%7B2%7D%5Csin %20%5Cfrac%7BC%7D%7B2%7D%7D%7D%7B%7B%5Ccos%20%5Cfr ac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D%7D%20%5Cright%29%20=4R%5Cc os%20%5Cfrac%7BB%7D%7B2%7D.%5Ccos%20%5Cfrac%7BC%7D %7B2%7D.%5Cfrac%7B%7B%5Csin%20%5Cfrac%7BA%7D%7B2%7 D%7D%7D%7B%7B%5Ccos%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D
[Only registered and activated users can see links] %5Ccos%20%5Cfrac%7BB%7D%7B2%7D%5Ccos%20%5Cfrac%7BC %7D%7B2%7Dtg%5Cfrac%7BA%7D%7B2%7D
Tương tự :
[Only registered and activated users can see links] c%7BA%7D%7B2%7D%5Ccos%20%5Cfrac%7BB%7D%7B2%7D%5Cco s%5Cfrac%7BC%7D%7B2%7Dtg%5Cfrac%7BB%7D%7B2%7D%7B%5 Crm%7B%20;%20C%7D%7D_%7B%5Crm%7B2%7D%7D%20C_0%20%2 0%5Cge%204R%5Ccos%5Cfrac%7BA%7D%7B2%7D%5Ccos%20%5C frac%7BB%7D%7B2%7D%5Ccos%20%5Cfrac%7BC%7D%7B2%7Dtg %5Cfrac%7BC%7D%7B2%7D
[Only registered and activated users can see links] 2%20B_0%20%20+%20C_2%20C_0%20%20%5Cge%204R%5Ccos%5 Cfrac%7BA%7D%7B2%7D%5Ccos%20%5Cfrac%7BB%7D%7B2%7D% 5Ccos%20%5Cfrac%7BC%7D%7B2%7D%5Cleft%28%20%7Btg%5C frac%7BA%7D%7B2%7D%20+tg%5Cfrac%7BB%7D%7B2%7D%20+% 20tg%5Cfrac%7BC%7D%7B2%7D%7D%20%5Cright%29
[Only registered and activated users can see links] rac%7BA%7D%7B2%7D%5Ccos%20%5Cfrac%7BB%7D%7B2%7D%5C cos%20%5Cfrac%7BC%7D%7B2%7D%20=
[Only registered and activated users can see links] 20%5Csin%20B%20+%20%5Csin%20C%29%20=%20%5Cfrac%7B% 7B%5Csqrt%203%20%7D%7D%7B2%7D%28a%20+%20b+%20c%29% 20=%20%5Csqrt%203%20p đpcm
Bài toán 5:Chứng minh rằng: a) file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image004.gif (Vô địch toán Quốc tế lần 30)
b) file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image006.gif
Lời giải :
a)Ta có: file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image008.gif . Tương tự: file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image010.gif; file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image012.gif
file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image014.gif
file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image016.gif (đpcm)
b)file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image018.gif
Tương tự :file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image020.giftheo tỉ số đồng dạng file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image022.gif file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image024.gif đpcm .
Từ bài toán 5 ta có được: file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image026.gif
CÁC BẤT ĐẲNG THỨC LIÊN QUAN ĐẾN PHAN GIÁC
Cho [Only registered and activated users can see links] Gọi Ao là tâm đường tròn bàng tiếp góc A, ; A1 và A2 lần lượt giao điểm của phân giác trong góc A với đường tròn ngoại tiếp tam giác ABC và cạnh BC .Các điểm Bi , Ci (i=0,1,2) được định nghĩa tương tự .Gọi I là tâm đường tròn nội tiếp tam giác ABC. Khi đó ta có các tính chất sau:
Tính chất 1: [Only registered and activated users can see links] %5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7Bb%20+%20c%7D%7D; [Only registered and activated users can see links] %5Cfrac%7BB%7D%7B2%7D%7D%7D%7B%7Bc%20+%20a%7D%7D ; [Only registered and activated users can see links] %5Cfrac%7BC%7D%7B2%7D%7D%7D%7B%7Ba%20+%20b%7D%7D
Tính chất 2: [Only registered and activated users can see links] 7D%7B%7B%5Ccos%20%5Cfrac%7BA%7D%7B2%7D%7D%7D ; [Only registered and activated users can see links] 7D%7B%7B%5Ccos%20%5Cfrac%7BB%7D%7B2%7D%7D%7D ; [Only registered and activated users can see links] 7D%7B%7B%5Ccos%20%5Cfrac%7BC%7D%7B2%7D%7D%7D
Tính chất 3: [Only registered and activated users can see links] 7B2%7D%5Ccos%20%5Cfrac%7BC%7D%7B2%7D ; [Only registered and activated users can see links] 7B2%7D%5Ccos%20%5Cfrac%7BA%7D%7B2%7D ; [Only registered and activated users can see links] 7B2%7D%5Ccos%20%5Cfrac%7BB%7D%7B2%7D
Tính chất 4: [Only registered and activated users can see links] D%5Csin%20%5Cfrac%7BC%7D%7B2%7D ; [Only registered and activated users can see links] D%5Csin%20%5Cfrac%7BA%7D%7B2%7D ; [Only registered and activated users can see links] D%5Csin%20%5Cfrac%7BB%7D%7B2%7D
[Only registered and activated users can see links]
Việc chứng minh các tính chất này xin dành cho bạn đọc
Bây giờ ta xét một số bài toán sau:
Bài toán 1: Ñaët [Only registered and activated users can see links] %7B%5Crm%7B%20=%20%7D%7D%5Cfrac%7B%7B%7B%5Crm%7BAA %7D%7D_%7B%5Crm%7B2%7D%7D%7D%7D%7B%7B%7B%5Crm%7BAA %7D%7D_%7B%5Crm%7B1%7D%7D%20%7D%7D ; [Only registered and activated users can see links] %7B%5Crm%7B%20=%20%7D%7D%5Cfrac%7B%7B%7B%5Crm%7BBB %7D%7D_%7B%5Crm%7B2%7D%7D%7D%7D%7B%7B%7B%5Crm%7BBB %7D%7D_%7B%5Crm%7B1%7D%7D%20%7D%7D ; [Only registered and activated users can see links] %7B%5Crm%7B%20=%20%7D%7D%5Cfrac%7B%7B%7B%5Crm%7BCC %7D%7D_%7B%5Crm%7B2%7D%7D%7D%7D%7B%7B%7B%5Crm%7BCC %7D%7D_%7B%5Crm%7B1%7D%7D%20%7D%7D
CMR: a) [Only registered and activated users can see links] %20%5Cfrac%7B9%7D%7B4%7D ;
b) [Only registered and activated users can see links] 5Cfrac%7B1%7D%7B%7Bl_b%20%7D%7D%20+%20%5Cfrac%7B1% 7D%7B%7Bl_c%20%7D%7D%20%5Cge%204
c/[Only registered and activated users can see links] %5E2%20A%7D%7D%20+%20%5Cfrac%7B%7Bl_b%20%7D%7D%7B% 7B%5Csin%20%5E2%20B%7D%7D%20+%20%5Cfrac%7B%7Bl_c%7 D%7D%7B%7B%5Csin%20%5E2%20C%7D%7D%20%5Cge%203 (Olympic chaâu AÙ 1998)
Lời giaûi:
a)Ta có : [Only registered and activated users can see links] 5CRightarrow [Only registered and activated users can see links] %7B%5Crm%7B%20=%20%7D%7D%5Cfrac%7B%7B%7B%5Crm%7B4b c%7D%7D%7B%5Crm%7B.cos%7D%7D%5E%7B%5Crm%7B2%7D%7D% 5Cfrac%7B%7B%5Crm%7BA%7D%7D%7D%7B%7B%5Crm%7B2%7D%7 D%7D%7D%7D%7B%7B%7B%5Crm%7B%28b%20+%20c%29%7D%7D%5 E%7B%5Crm%7B2%7D%7D%20%7D%7D%20%5Cle%7B%5Crm%7Bcos %7D%7D%5E%7B%5Crm%7B2%7D%7D%20%5Cfrac%7B%7B%5Crm%7 BA%7D%7D%7D%7B%7B%5Crm%7B2%7D%7D%7D chứng minh tương tự ta có: [Only registered and activated users can see links] %20%5Cle%20%7B%5Crm%7Bcos%7D%7D%5E%7B%5Crm%7B2%7D% 7D%5Cfrac%7B%7B%5Crm%7BB%7D%7D%7D%7B%7B%5Crm%7B2%7 D%7D%7D;%7B%5Crm%7B%20l%7D%7D_%7B%5Crm%7Bb%7D%7D%2 0%20%5Cle%20%7B%5Crm%7Bcos%7D%7D%5E%7B%5Crm%7B2%7D %7D%5Cfrac%7B%7B%5Crm%7BC%7D%7D%7D%7B%7B%5Crm%7B2% 7D%7D%7D
[Only registered and activated users can see links] 0+%20l_c%20%20%5Cle%20%5Ccos%20%5E2%20%5Cfrac%7BA% 7D%7B2%7D%20+%20%5Ccos%20%5E2%5Cfrac%7BB%7D%7B2%7D %20+%20%5Ccos%20%5E2%20%5Cfrac%7BC%7D%7B2%7D%20%5C le%20%5Cfrac%7B9%7D%7B4%7D
b/ Ta có : [Only registered and activated users can see links] %20%20+%20%7B%5Crm%7Bl%7D%7D_%7B%5Crm%7Bb%7D%7D%20 %20+%20%7B%5Crm%7Bl%7D%7D_%7B%5Crm%7Bc%7D%7D%7B%5C rm%7B%29%28%7D%7D%5Cfrac%7B%7B%5Crm%7B1%7D%7D%7D%7 B%7B%7B%5Crm%7Bl%7D%7D_%7B%5Crm%7Ba%7D%7D%20%7D%7D %20+%5Cfrac%7B%7B%5Crm%7B1%7D%7D%7D%7B%7B%7B%5Crm% 7Bl%7D%7D_%7B%5Crm%7Bb%7D%7D%20%7D%7D%20+%5Cfrac%7 B%7B%5Crm%7B1%7D%7D%7D%7B%7B%7B%5Crm%7Bl%7D%7D_%7B %5Crm%7Bc%7D%7D%20%7D%7D%7B%5Crm%7B%29%7D%7D%20%5C ge%209
[Only registered and activated users can see links] _a%20%7D%7D%20+%20%5Cfrac%7B1%7D%7B%7Bl_b%20%7D%7D %20+%20%5Cfrac%7B1%7D%7B%7Bl_c%20%7D%7D%5Cge%204
c)Ta có:
[Only registered and activated users can see links] 5E2%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B%28b%20+%20 c%29%5E2%20%7D%7D%20=%5Cfrac%7B%7B4.2R%5Csin%20B.2 R%5Csin%20C.%5Ccos%20%5E2%20%5Cfrac%7BA%7D%7B2%7D% 7D%7D%7B%7B%282R%5Csin%20B%20+%202R%5CsinC%29%5E2% 20%7D%7D%20=%20%5Cfrac%7B%7B4%5Csin%20B.%5Csin%20C .%5Ccos%20%5E2%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B %28%5Csin%20B%20+%20%5CsinC%29%5E2%20%7D%7D%20=%20 %5Cfrac%7B%7B4%5Csin%20B.%5Csin%20C.%5Ccos%20%5E2% 20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B4%5Ccos%20%5E2% 5Cfrac%7BA%7D%7B2%7D.%5Ccos%20%5E2%20%5Cfrac%7B%7B B%20-%20C%7D%7D%7B2%7D%7D%7D
[Only registered and activated users can see links] 7B%7B%5Csin%20B.%5Csin%20C%7D%7D%7B%7B%5Ccos%20%5E 2%20%5Cfrac%7B%7BB%20-C%7D%7D%7B2%7D%7D%7D%20%5Cge%20%5Csin%20B.%5Csin%2 0C%20%5CRightarrow%20%5Cfrac%7B%7Bl_a%20%7D%7D%7B% 7B%5Csin%20%5E2%20A%7D%7D%20%5Cge%5Cfrac%7B%7B%5Cs in%20B.%5Csin%20C%7D%7D%7B%7B%5Csin%20%5E2%20A%7D% 7D .
Chứng minh tương tự ta cũng có
[Only registered and activated users can see links] %5E2%20B%7D%7D%20%5Cge%20%5Cfrac%7B%7B%5Csin%20C.% 5Csin%20A%7D%7D%7B%7B%5Csin%20%5E2B%7D%7D%7B%5Crm% 7B%20;%20%20%20%20%7D%7D%5Cfrac%7B%7B%7B%5Crm%7Bl% 7D%7D_%7B%5Crm%7Bc%7D%7D%20%7D%7D%7B%7B%7B%5Crm%7B sin%7D%7D%5E%7B%5Crm%7B2%7D%7D%20C%7D%7D%5Cge%20%5 Cfrac%7B%7B%5Csin%20A.%5Csin%20B%7D%7D%7B%7B%5Csin %20%5E2%20C%7D%7D
[Only registered and activated users can see links] D%7D%7B%7B%5Csin%20%5E2%20A%7D%7D%20+%20%5Cfrac%7B %7Bl_b%20%7D%7D%7B%7B%5Csin%20%5E2%20B%7D%7D+%20%5 Cfrac%7B%7Bl_c%20%7D%7D%7B%7B%5Csin%20%5E2%20C%7D% 7D%20%5Cge%20%5Cfrac%7B%7B%5Csin%20B.%5Csin%20C%7D %7D%7B%7B%5Csin%20%5E2%20A%7D%7D%20+%5Cfrac%7B%7B% 5Csin%20C.%5Csin%20A%7D%7D%7B%7B%5Csin%20%5E2%20B% 7D%7D%20+%20%5Cfrac%7B%7B%5Csin%20A.%5Csin%20B%7D% 7D%7B%7B%5Csin%20%5E2C%7D%7D%20%5Cge%203%5Csqrt[3]%7B%7B%5Cfrac%7B%7B%5Csin%20%5E2%20%7B%5Crm%7BA%7D %7D%7B%5Crm%7B.sin%7D%7D%5E%7B%5Crm%7B2%7D%7D%20B. %5Csin%5E2%20C%7D%7D%7B%7B%5Csin%20%5E2%20%7B%5Crm %7BA%7D%7D%7B%5Crm%7B.sin%7D%7D%5E%7B%5Crm%7B2%7D% 7D%20B.%5Csin%20%5E2%20C%7D%7D%7D%7D%20=%203
Đẳng thức xảy ra khi và chỉ khi A = B = C
Bài toán 2 : Đặt [Only registered and activated users can see links] 7B%7BAA_0%20%7D%7D;%7B%5Crm%7B%20%7D%7Dx_b%20%20=% 20%5Cfrac%7B%7BBB_2%20%7D%7D%7B%7BBB_0%7D%7D;%7B%5 Crm%7B%20%7D%7Dx_c%20%20=%20%5Cfrac%7B%7BCC_2%20%7 D%7D%7B%7BCC_0%20%7D%7D. Cmr:
a) [Only registered and activated users can see links] 20+%20x_c%20%20%5Cle%20%5Cfrac%7B3%7D%7B2%7D
[Only registered and activated users can see links] rm%7B1%7D%7D%7D%7B%7B%7B%5Crm%7Bx%7D%7D_%7B%5Crm%7 Ba%7D%7D%20%7D%7D%20+%20%5Cfrac%7B1%7D%7B%7Bx_b%20 %7D%7D%20+%5Cfrac%7B1%7D%7B%7Bx_c%20%7D%7D%20%5Cge %206%7B%5Crm%7B%20%7D%7D
[Only registered and activated users can see links] %7B%5Crm%7Bx%7D%7D_%7B%5Crm%7Ba%7D%7D%20%7D%7D%7B% 7B%5Csin%20%5E2%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20+% 5Cfrac%7B%7B%7B%5Crm%7Bx%7D%7D_%7B%5Crm%7Bb%7D%7D% 20%7D%7D%7B%7B%5Csin%20%5E2%20%5Cfrac%7BB%7D%7B2%7 D%7D%7D%20+%5Cfrac%7B%7B%7B%5Crm%7Bx%7D%7D_%7B%5Cr m%7Bc%7D%7D%20%7D%7D%7B%7B%5Csin%20%5E2%20%5Cfrac% 7BC%7D%7B2%7D%7D%7D%20%5Cge%206
Lời giải:
a) Ta có: [Only registered and activated users can see links] 7B%7BAA_0%20%7D%7D%20=%20%5Cfrac%7B%7B2bc.%5Ccos%2 0%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B4R%28b+%20c%29.% 5Ccos%20%5Cfrac%7BB%7D%7B2%7D.%5Ccos%20%5Cfrac%7BC %7D%7B2%7D%7D%7D%20=%20%5Cfrac%7B%7B%5Csin%20B.%5C sin%20C.%5Ccos%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B%28 %5Csin%20B%20+%20%5Csin%20C%29.%5Ccos%20%5Cfrac%7B B%7D%7B2%7D.%5Ccos%20%5Cfrac%7BC%7D%7B2%7D%7D%7D%2 0=
[Only registered and activated users can see links] B%7D%7B2%7D.%5Csin%20%5Cfrac%7BC%7D%7B2%7D.%5Ccos% 20%5Cfrac%7BB%7D%7B2%7D.%5Ccos%5Cfrac%7BC%7D%7B2%7 D.%5Ccos%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B2%5Cco s%20%5Cfrac%7BA%7D%7B2%7D.%5Ccos%20%5Cfrac%7B%7BB% 20-C%7D%7D%7B2%7D.%5Ccos%20%5Cfrac%7BB%7D%7B2%7D.%5Cc os%20%5Cfrac%7BC%7D%7B2%7D%7D%7D%20=%20%5Cfrac%7B% 7B2%5Csin%5Cfrac%7BB%7D%7B2%7D.%5Csin%20%5Cfrac%7B C%7D%7B2%7D%7D%7D%7B%7B%5Ccos%20%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D
[Only registered and activated users can see links] 7BB%20-%20C%7D%7D%7B2%7D%20-%20%5Csin%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B%7B%5Cco s%20%5Cfrac%7B%7BB-%20C%7D%7D%7B2%7D%7D%7D%20=%201%20-%20%5Cfrac%7B%7B%5Csin%20%5Cfrac%7BA%7D%7B2%7D%7D% 7D%7B%7B%5Ccos%20%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D
Tương tự: [Only registered and activated users can see links] 7D%7B%7B%5Ccos%20%5Cfrac%7B%7BC%20-%20A%7D%7D%7B2%7D%7D%7D%7B%5Crm%7B%20%7D%7D;%7B%5C rm%7B%20%20%20%20x%7D%7D_%7B%5Crm%7Bc%7D%7D%20%7B% 5Crm%7B%20=%20%7D%7D1%20-%5Cfrac%7B%7B%7B%5Crm%7Bsin%7D%7D%5Cfrac%7B%7B%5Cr m%7BC%7D%7D%7D%7B%7B%5Crm%7B2%7D%7D%7D%7D%7D%7B%7B %5Ccos%20%5Cfrac%7B%7BA%20-B%7D%7D%7B2%7D%7D%7D%7B%5Crm%7B%20%7D%7D
[Only registered and activated users can see links] 0+%20x_c%20%20=%203%20-%20%28%5Cfrac%7B%7B%5Csin%5Cfrac%7BA%7D%7B2%7D%7D% 7D%7B%7B%5Ccos%20%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B%7B%5Csin% 5Cfrac%7BB%7D%7B2%7D%7D%7D%7B%7B%5Ccos%20%5Cfrac%7 B%7BC%20-%20A%7D%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B%7B%5Csin% 5Cfrac%7BC%7D%7B2%7D%7D%7D%7B%7B%5Ccos%20%5Cfrac%7 B%7BA%20-%20B%7D%7D%7B2%7D%7D%7D%29
Mặt khác: [Only registered and activated users can see links] %7D%7D%7D%7B%7B%5Ccos%20%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B%7B%5Csin% 5Cfrac%7BB%7D%7B2%7D%7D%7D%7B%7B%5Ccos%20%5Cfrac%7 B%7BC%20-%20A%7D%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B%7B%5Csin% 5Cfrac%7BC%7D%7B2%7D%7D%7D%7B%7B%5Ccos%20%5Cfrac%7 B%7BA%20-%20B%7D%7D%7B2%7D%7D%7D
[Only registered and activated users can see links] 7B%5Csin%20B%20+%20%5Csin%20C%7D%7D%20+%20%5Cfrac% 7B%7B%5Csin%20B%7D%7D%7B%7B%5Csin%20C%20+%5Csin%20 A%7D%7D%20+%20%5Cfrac%7B%7B%5Csin%20C%7D%7D%7B%7B% 5Csin%20A%20+%20%5Csin%20B%7D%7D%20%5Cge%20%5Cfrac %7B3%7D%7B2%7D
[Only registered and activated users can see links] 0+%20x_c%20%20%5Cle%203%20-%20%5Cfrac%7B3%7D%7B2%7D%20=%20%5Cfrac%7B3%7D%7B2% 7D. Đẳng thức xảy ra khi và chỉ khi tam giác ABC đều
b)Từ Bđt : [Only registered and activated users can see links] %20%20+%20x_%7B%5Crm%7Bb%7D%7D%20%20+%20x_%7B%5Crm %7Bc%7D%7D%7B%5Crm%7B%29%28%7D%7D%5Cfrac%7B%7B%5Cr m%7B1%7D%7D%7D%7B%7Bx_%7B%5Crm%7Ba%7D%7D%20%7D%7D% 20+%20%5Cfrac%7B%7B%5Crm%7B1%7D%7D%7D%7B%7Bx_%7B%5 Crm%7Bb%7D%7D%7D%7D%20+%20%5Cfrac%7B%7B%5Crm%7B1%7 D%7D%7D%7B%7Bx_%7B%5Crm%7Bc%7D%7D%20%7D%7D%7B%5Crm %7B%29%7D%7D%20%5Cge%209%20%5CRightarrow%5Cfrac%7B %7B%5Crm%7B1%7D%7D%7D%7B%7Bx_%7B%5Crm%7Ba%7D%7D%20 %7D%7D%20+%20%5Cfrac%7B%7B%5Crm%7B1%7D%7D%7D%7B%7B x_%7B%5Crm%7Bb%7D%7D%20%7D%7D%20+%5Cfrac%7B%7B%5Cr m%7B1%7D%7D%7D%7B%7Bx_%7B%5Crm%7Bc%7D%7D%20%7D%7D% 20%5Cge%20%5Cfrac%7B9%7D%7B%7B%7B%5Crm%7Bx%7D%7D_% 7B%5Crm%7Ba%7D%7D%20%20+x_%7B%5Crm%7Bb%7D%7D%20%20 +%20x_%7B%5Crm%7Bc%7D%7D%20%7D%7D%20%5Cge%206 (đpcm)
c)Ta có: [Only registered and activated users can see links] rac%7BB%7D%7B2%7D.%5Csin%20%5Cfrac%7BC%7D%7B2%7D%7 D%7D%7B%7B%5Ccos%20%5Cfrac%7B%7BB%20-C%7D%7D%7B2%7D%7D%7D%20%5Cge%202%5Csin%20%5Cfrac%7 BB%7D%7B2%7D.%5Csin%20%5Cfrac%7BC%7D%7B2%7D%20%5CR ightarrow%20%5Cfrac%7B%7Bx_a%7D%7D%7B%7B%5Csin%20% 5E2%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20%5Cge%202%5Cfr ac%7B%7B%5Csin%20%5Cfrac%7BB%7D%7B2%7D.%5Csin%5Cfr ac%7BC%7D%7B2%7D%7D%7D%7B%7B%5Csin%20%5E2%20%5Cfra c%7BA%7D%7B2%7D%7D%7D.
Chứng minh tương tự ta có:
[Only registered and activated users can see links] %5E2%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20%5Cge%202%5Cf rac%7B%7B%5Csin%5Cfrac%7BC%7D%7B2%7D.%5Csin%20%5Cf rac%7BA%7D%7B2%7D%7D%7D%7B%7B%5Csin%20%5E2%20%5Cfr ac%7BB%7D%7B2%7D%7D%7D%7B%5Crm%7B%20%20;%20%7D%7D% 5Cfrac%7B%7Bx_c%20%7D%7D%7B%7B%5Csin%20%5E2%20%5Cf rac%7BC%7D%7B2%7D%7D%7D%20%5Cge%202%5Cfrac%7B%7B%5 Csin%5Cfrac%7BA%7D%7B2%7D.%5Csin%20%5Cfrac%7BB%7D% 7B2%7D%7D%7D%7B%7B%5Csin%20%5E2%20%5Cfrac%7BC%7D%7 B2%7D%7D%7D
[Only registered and activated users can see links] D%7D%7B%7B%5Csin%20%5E2%20%5Cfrac%7BA%7D%7B2%7D%7D %7D%20+%20%5Cfrac%7B%7Bx_b%7D%7D%7B%7B%5Csin%20%5E 2%20%5Cfrac%7BB%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B%7 Bx_c%20%7D%7D%7B%7B%5Csin%20%5E2%20%5Cfrac%7BC%7D% 7B2%7D%7D%7D
[Only registered and activated users can see links] Csin%20%5Cfrac%7BB%7D%7B2%7D.%5Csin%20%5Cfrac%7BC% 7D%7B2%7D%7D%7D%7B%7B%5Csin%20%5E2%5Cfrac%7BA%7D%7 B2%7D%7D%7D%20+%20%5Cfrac%7B%7B%5Csin%20%5Cfrac%7B C%7D%7B2%7D.%5Csin%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7 B%7B%5Csin%20%5E2%5Cfrac%7BB%7D%7B2%7D%7D%7D%20+%2 0%5Cfrac%7B%7B%5Csin%20%5Cfrac%7BA%7D%7B2%7D.%5Csi n%20%5Cfrac%7BB%7D%7B2%7D%7D%7D%7B%7B%5Csin%20%5E2 %5Cfrac%7BC%7D%7B2%7D%7D%7D%7D%20%5Cright%29%20%5C ge%202.3%20=%206.
Bài 3: Chứng minh rằng
[Only registered and activated users can see links] 20%20%5Cge%20IA%20+%20IB%20+%20IC
[Only registered and activated users can see links] D%7B%7BIA%7D%7D%20+%20%5Cfrac%7B1%7D%7B%7BIB%7D%7D %20+%20%5Cfrac%7B1%7D%7B%7BIC%7D%7D%20%5Cge%5Cfrac %7B1%7D%7B%7BIA_1%20%7D%7D%20+%20%5Cfrac%7B1%7D%7B %7BIB_1%20%7D%7D%20+%20%5Cfrac%7B1%7D%7B%7BIC_1%20 %7D%7D%20%5Cge%5Cfrac%7B3%7D%7BR%7D%20%5C%5C%20%20 %7B%5Crm%7B%20%20%20%7D%7D%20%5C%5C%20%20%5Cend%7B array%7D
Lời giải : a) Ta có: [Only registered and activated users can see links] cos%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20-%204R%5Csin%5Cfrac%7BB%7D%7B2%7D%5Csin%20%5Cfrac%7 BC%7D%7B2%7D%20=%20%5Cfrac%7B%7BR%28%5Csin%20B%20+ %20%5Csin%20C%29%7D%7D%7B%7B%5Ccos%5Cfrac%7BA%7D%7 B2%7D%7D%7D%20-%204R%5Csin%20%5Cfrac%7BB%7D%7B2%7D%5Csin%20%5Cfra c%7BC%7D%7B2%7D
[Only registered and activated users can see links] c%7BC%7D%7B2%7D%20=2R%5Csin%20%5Cfrac%7BA%7D%7B2%7 D%7B%5Crm%7B%20%20%20%20%7D%7DT%5C
[Only registered and activated users can see links] %20+%20IC_1%20%20=%202R%28%5Csin%20%5Cfrac%7BA%7D% 7B2%7D%20+%20%5Csin%5Cfrac%7BB%7D%7B2%7D%20+%20%5C sin%20%5Cfrac%7BC%7D%7B2%7D%29
[Only registered and activated users can see links]
[Only registered and activated users can see links] %20%20%7D%7D%20%5Cge%202%28%5Csin%20%5Cfrac%7BA%7D %7B2%7D.%5Csin%20%5Cfrac%7BA%7D%7B2%7D%20+%20%5Csi n%5Cfrac%7BB%7D%7B2%7D.%5Csin%20%5Cfrac%7BC%7D%7B2 %7D%20+%20%5Csin%20%5Cfrac%7BC%7D%7B2%7D.%5Csin%20 %5Cfrac%7BA%7D%7B2%7D%29%20%5C%5C%20%20%5Cend%7Bar ray%7D
[Only registered and activated users can see links] %20+%20IC_1%20%20%5Cge%204R%28%5Csin%20%5Cfrac%7BA %7D%7B2%7D%5Csin%5Cfrac%7BB%7D%7B2%7D%20+%20%5Csin %20%5Cfrac%7BB%7D%7B2%7D%5Csin%20%5Cfrac%7BC%7D%7B 2%7D%20+%20%5Csin%20%5Cfrac%7BC%7D%7B2%7D%5Csin%5C frac%7BA%7D%7B2%7D%29%20=%20IA%20+%20IB%20+%20IC đpcm
[Only registered and activated users can see links] 7D%7D%5Cfrac%7B1%7D%7B%7BIA%7D%7D%20+%20%5Cfrac%7B 1%7D%7B%7BIB%7D%7D%20+%5Cfrac%7B1%7D%7B%7BIC%7D%7D %20=%20%5Cfrac%7B1%7D%7B%7B4R%7D%7D%28%5Cfrac%7B1% 7D%7B%7B%5Csin%20%5Cfrac%7BA%7D%7B2%7D%5Csin%5Cfra c%7BB%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B1%7D%7B%7B%5 Csin%20%5Cfrac%7BB%7D%7B2%7D%5Csin%20%5Cfrac%7BC%7 D%7B2%7D%7D%7D%20+%5Cfrac%7B1%7D%7B%7B%5Csin%20%5C frac%7BC%7D%7B2%7D%5Csin%20%5Cfrac%7BA%7D%7B2%7D%7 D%7D%29%20=%20%20%5C%5C
[Only registered and activated users can see links] Cfrac%7B%7B%5Csin%20%5Cfrac%7BA%7D%7B2%7D%20+%20%5 Csin%20%5Cfrac%7BB%7D%7B2%7D%20+%20%5Csin%5Cfrac%7 BC%7D%7B2%7D%7D%7D%7B%7B%5Csin%20%5Cfrac%7BA%7D%7B 2%7D%5Csin%20%5Cfrac%7BB%7D%7B2%7D%5Csin%20%5Cfrac %7BC%7D%7B2%7D%7D%7D%20%5Cge%5Cfrac%7B1%7D%7B%7B4R %7D%7D.%5Cfrac%7B%7B2%28%5Csin%20%5Cfrac%7BA%7D%7B 2%7D.%5Csin%20%5Cfrac%7BA%7D%7B2%7D%20+%20%5Csin%5 Cfrac%7BB%7D%7B2%7D.%5Csin%20%5Cfrac%7BC%7D%7B2%7D %20+%20%5Csin%20%5Cfrac%7BC%7D%7B2%7D.%5Csin%5Cfra c%7BA%7D%7B2%7D%29%7D%7D%7B%7B%5Csin%20%5Cfrac%7BA %7D%7B2%7D%5Csin%20%5Cfrac%7BB%7D%7B2%7D%5Csin%20% 5Cfrac%7BC%7D%7B2%7D%7D%7D%20=
[Only registered and activated users can see links] 0%5Cfrac%7BA%7D%7B2%7D%7D%7D%20+%20%5Cfrac%7B1%7D% 7B%7B2R%5Csin%20%5Cfrac%7BB%7D%7B2%7D%7D%7D+%20%5C frac%7B1%7D%7B%7B2R%5Csin%20%5Cfrac%7BC%7D%7B2%7D% 7D%7D%20=%20%5Cfrac%7B1%7D%7B%7BIA_1%20%7D%7D%20+% 20%5Cfrac%7B1%7D%7B%7BIB_1%7D%7D%20+%20%5Cfrac%7B1 %7D%7B%7BIC_1%20%7D%7D%20%5C%5C%20%20%5Cend%7Barra y%7D
Bài toán 4:Chứng minh rằng:
[Only registered and activated users can see links] %20B_1%20B_2%20%20+%20C_1%20C_2%20%20%5Cge%20%5Cfr ac%7B%7B2p%5E2%20%7D%7D%7B%7B9R%7D%7D
[Only registered and activated users can see links] 0+%20C_2%20C_0%20%20%5Cge%20%5Csqrt%203%20p
a)Ta có : [Only registered and activated users can see links] B2%5Ccos%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20-%5Cfrac%7B%7B2bc.%5Ccos%20%5Cfrac%7BA%7D%7B2%7D%7D %7D%7B%7Bb%20+%20c%7D%7D%20=%20%5Cfrac%7B%7B%28b%2 0+%20c%29%5E2%20%20-%202bc.%5Ccos%5E2%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%7B %7B2%28b%20+%20c%29%5Ccos%20%5Cfrac%7BA%7D%7B2%7D% 7D%7D%20=
[Only registered and activated users can see links] 5E2%20%20-%204bc.%5Ccos%20A%7D%7D%7B%7B2%28b%20+%20c%29%5Cco s%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20=%5Cfrac%7B%7Ba% 5E2%20%7D%7D%7B%7B8R%28%5Csin%20B%20+%20%5Csin%20C %29%5Ccos%20%5Cfrac%7BA%7D%7B2%7D%7D%7D%20=%20%5Cf rac%7B%7Ba%5E2%7D%7D%7B%7B8R.%5Ccos%20%5E2%20%5Cfr ac%7BA%7D%7B2%7D.%5Ccos%20%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D%20%5Cge%20%5Cfrac%7B%7Ba%5 E2%7D%7D%7B%7B8R.%5Ccos%20%5E2%20%5Cfrac%7BA%7D%7B 2%7D%7D%7D
Tương tự[Only registered and activated users can see links] 20%5Cge%20%5Cfrac%7B%7Bb%5E2%20%7D%7D%7B%7B8R.%5Cc os%20%5E2%20%5Cfrac%7BB%7D%7B2%7D%7D%7D%7B%5Crm%7B %20%7D%7D;%7B%5Crm%7B%20%20C%7D%7D_%7B%5Crm%7B1%7D %7D%20%7B%5Crm%7BC%7D%7D_%7B%5Crm%7B2%7D%7D%20%20% 5Cge%5Cfrac%7B%7B%7B%5Crm%7Bc%7D%7D%5E%7B%5Crm%7B2 %7D%7D%20%7D%7D%7B%7B8R.%5Ccos%20%5E2%20%5Cfrac%7B C%7D%7B2%7D%7D%7D%7B%5Crm%7B%20%7D%7D
[Only registered and activated users can see links] 1%20B_2%20%20+%20C_1%20C_2%20%20%5Cge%20%5Cfrac%7B 1%7D%7B%7B8R%7D%7D%5Cleft%28%7B%5Cfrac%7B%7Ba%5E2% 20%7D%7D%7B%7B%5Ccos%20%5E2%20%5Cfrac%7BA%7D%7B2%7 D%7D%7D%20+%20%5Cfrac%7B%7Bb%5E2%20%7D%7D%7B%7B%5C cos%20%5E2%5Cfrac%7BB%7D%7B2%7D%7D%7D%20+%20%5Cfra c%7B%7Bc%5E2%20%7D%7D%7B%7B%5Ccos%20%5E2%20%5Cfrac %7BC%7D%7B2%7D%7D%7D%7D%20%5Cright%29
Mặt khác áp dụng Bđt Bunhia ta có:
[Only registered and activated users can see links] 7BA%7D%7B2%7D%20+%20%5Ccos%20%5E2%20%5Cfrac%7BB%7D %7B2%7D%20+%20%5Ccos%20%5E2%5Cfrac%7BC%7D%7B2%7D%7 D%20%5Cright%29%5Cleft%28%20%7B%5Cfrac%7B%7Ba%5E2% 20%7D%7D%7B%7B%5Ccos%20%5E2%20%5Cfrac%7BA%7D%7B2%7 D%7D%7D%20+%5Cfrac%7B%7Bb%5E2%20%7D%7D%7B%7B%5Ccos %20%5E2%20%5Cfrac%7BB%7D%7B2%7D%7D%7D%20+%20%5Cfra c%7B%7Bc%5E2%20%7D%7D%7B%7B%5Ccos%20%5E2%5Cfrac%7B C%7D%7B2%7D%7D%7D%7D%20%5Cright%29%20%5Cge%20%28a% 20+%20b%20+%20c%29%5E2
[Only registered and activated users can see links] 1%20B_2%20%20+%20C_1%20C_2%20%20%5Cge%5Cfrac%7B1%7 D%7B%7B8R%7D%7D%5Cfrac%7B%7B%28a%20+%20b%20+%20c%2 9%5E2%20%7D%7D%7B%7B%5Ccos%20%5E2%20%5Cfrac%7BA%7D %7B2%7D%20+%20%5Ccos%20%5E2%5Cfrac%7BB%7D%7B2%7D%2 0+%20%5Ccos%20%5E2%20%5Cfrac%7BC%7D%7B2%7D%7D%7D%2 0%5Cge%20%5Cfrac%7B%7B4p%5E2%7D%7D%7B%7B8R.%5Cfrac %7B9%7D%7B4%7D%7D%7D%20=%20%5Cfrac%7B%7B2p%5E2%20% 7D%7D%7B%7B9R%7D%7D đpcm
b)Ta có:
[Only registered and activated users can see links] %5Ccos%20%5Cfrac%7BC%7D%7B2%7D%20-%5Cfrac%7B%7B2bc.%5Ccos%20%5Cfrac%7BA%7D%7B2%7D%7D %7D%7B%7Bb%20+%20c%7D%7D%20=%204R%5Ccos%20%5Cfrac% 7BB%7D%7B2%7D%5Ccos%5Cfrac%7BC%7D%7B2%7D%20-%20%5Cfrac%7B%7B4R%5Csin%20B%5Csin%20C%7D%7D%7B%7B 2%5Ccos%20%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D%20=
[Only registered and activated users can see links] 5Ccos%20%5Cfrac%7BC%7D%7B2%7D%5Cleft%28%20%7B1%20-%20%5Cfrac%7B%7B2%5Csin%5Cfrac%7BB%7D%7B2%7D%5Csin %20%5Cfrac%7BC%7D%7B2%7D%7D%7D%7B%7B%5Ccos%20%5Cfr ac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D%7D%20%5Cright%29%20=4R%5Cc os%20%5Cfrac%7BB%7D%7B2%7D.%5Ccos%20%5Cfrac%7BC%7D %7B2%7D.%5Cfrac%7B%7B%5Csin%20%5Cfrac%7BA%7D%7B2%7 D%7D%7D%7B%7B%5Ccos%5Cfrac%7B%7BB%20-%20C%7D%7D%7B2%7D%7D%7D
[Only registered and activated users can see links] %5Ccos%20%5Cfrac%7BB%7D%7B2%7D%5Ccos%20%5Cfrac%7BC %7D%7B2%7Dtg%5Cfrac%7BA%7D%7B2%7D
Tương tự :
[Only registered and activated users can see links] c%7BA%7D%7B2%7D%5Ccos%20%5Cfrac%7BB%7D%7B2%7D%5Cco s%5Cfrac%7BC%7D%7B2%7Dtg%5Cfrac%7BB%7D%7B2%7D%7B%5 Crm%7B%20;%20C%7D%7D_%7B%5Crm%7B2%7D%7D%20C_0%20%2 0%5Cge%204R%5Ccos%5Cfrac%7BA%7D%7B2%7D%5Ccos%20%5C frac%7BB%7D%7B2%7D%5Ccos%20%5Cfrac%7BC%7D%7B2%7Dtg %5Cfrac%7BC%7D%7B2%7D
[Only registered and activated users can see links] 2%20B_0%20%20+%20C_2%20C_0%20%20%5Cge%204R%5Ccos%5 Cfrac%7BA%7D%7B2%7D%5Ccos%20%5Cfrac%7BB%7D%7B2%7D% 5Ccos%20%5Cfrac%7BC%7D%7B2%7D%5Cleft%28%20%7Btg%5C frac%7BA%7D%7B2%7D%20+tg%5Cfrac%7BB%7D%7B2%7D%20+% 20tg%5Cfrac%7BC%7D%7B2%7D%7D%20%5Cright%29
[Only registered and activated users can see links] rac%7BA%7D%7B2%7D%5Ccos%20%5Cfrac%7BB%7D%7B2%7D%5C cos%20%5Cfrac%7BC%7D%7B2%7D%20=
[Only registered and activated users can see links] 20%5Csin%20B%20+%20%5Csin%20C%29%20=%20%5Cfrac%7B% 7B%5Csqrt%203%20%7D%7D%7B2%7D%28a%20+%20b+%20c%29% 20=%20%5Csqrt%203%20p đpcm
Bài toán 5:Chứng minh rằng: a) file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image004.gif (Vô địch toán Quốc tế lần 30)
b) file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image006.gif
Lời giải :
a)Ta có: file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image008.gif . Tương tự: file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image010.gif; file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image012.gif
file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image014.gif
file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image016.gif (đpcm)
b)file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image018.gif
Tương tự :file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image020.giftheo tỉ số đồng dạng file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image022.gif file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image024.gif đpcm .
Từ bài toán 5 ta có được: file:///C:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/msohtml1/01/clip_image026.gif