ThanhTam
27-04-2009, 05:05 PM
CÁC CHỨNG MINH KHÁC NHAU
CHO BẤT ĐẢNG THỨC ĐƠN GIẢN
Chứng minh rằng [Only registered and activated users can see links]
trong đó A, B, C là ba góc của một tam giác bất kì .
( Theo thứ tự chương trình học Phổ thông )
Cách 1 Dùng tỉ số Diện Tích
Kẻ các đường cao AD, BE, CF
Đặt [Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links] [Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
Tương tự
[Only registered and activated users can see links]
[Only registered and activated users can see links]
Cộng (1), (2), (3) ta có
[Only registered and activated users can see links]
[Only registered and activated users can see links](đpcm )
Đẳng thức xảy ra khi và chỉ khi tam giác ABC đều.
Cách 2: Vận dụng bất đẳng thức :Erdos-Mordell
Cho tam giác ABC. M là một điểm bất kì nằm trong tam giác .
Đặt [Only registered and activated users can see links]
và [Only registered and activated users can see links] lần lượt là khoảng cách từ M đến BC, CA, AB tương ứng.
Khi đó ta có bất đẳng thức [Only registered and activated users can see links]
Vận dụng giải bài trên:
Gọi O , R là tâm và bán kính đường tròn ngoại tiếp tam giác ABC.
Gọi M, N, P lần lượt là trung điểm của cạnh AB, BC, CA..
Ta dễ dàng nhận thấy [Only registered and activated users can see links]
Do đó :[Only registered and activated users can see links]
Tương tự [Only registered and activated users can see links]
Do đó [Only registered and activated users can see links]
[Only registered and activated users can see links] ( đpcm).(Erdos-Mordell)
Đẳng thức xảy ra khi và chỉ khi tam giác ABC đều.
Cách 3 Sử dụng BĐT Trêbưsep.
Gọi a, b, c là ba cạnh tam giác, sử dụng công thức hình chiếu ta có:
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
Cộng ba biểu thức trên ta có:
[Only registered and activated users can see links]
Không mất tính tổng quát giả sử: [Only registered and activated users can see links] ta có:
[Only registered and activated users can see links]
Do đó :[Only registered and activated users can see links]
[Only registered and activated users can see links] ( Trêbưsep)
[Only registered and activated users can see links] (đpcm)
Đẳng thức xảy ra khi tam giác ABC đều.
Cách 4 Phuong pháp vectơ.
Gọi I và r lần lượt là tâm và bán kính đường tròn nội tiếp tam giác ABC,
và M, N, P lần lượt là tiếp điểm của đường tròn đó
với các cạnh AB, AC, BC ,ta có
[Only registered and activated users can see links]
[Only registered and activated users can see links] (*)
Ta nhận thấy
[Only registered and activated users can see links]
( Vì [Only registered and activated users can see links] và góc A bù nhau)
Tương tự :[Only registered and activated users can see links] [Only registered and activated users can see links]
Vậy từ (*) suy ra [Only registered and activated users can see links] (dpcm)
Cách 5: Phuong pháp vectơ.
Lấy A, B, C lần lượt là ba gốc của ba véctơ đơn vị sau
[Only registered and activated users can see links] [Only registered and activated users can see links]
[Only registered and activated users can see links]
Ta có :[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links] [Only registered and activated users can see links]
Cách 6: Quan hệ bất đẳng thức Schur.
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]( Schur)
Cách 7 Sử dụng tam thức bậc hai.
Xét [Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
Đặt [Only registered and activated users can see links]
Xét tam thức [Only registered and activated users can see links]
Có [Only registered and activated users can see links]
và hệ số [Only registered and activated users can see links]ên [Only registered and activated users can see links] với mọi x
Hay [Only registered and activated users can see links]
Cách 8: Sử dụng hàm số.
Ta có [Only registered and activated users can see links]
[Only registered and activated users can see links]
Đặt [Only registered and activated users can see links]
điều kiện [Only registered and activated users can see links]ét hàm số [Only registered and activated users can see links]
Lập bảng xét dấu ta có [Only registered and activated users can see links]
Cách 9 Tổng bình phương.
Xét [Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links] (Đúng)
Đẳng thức xảy ra khi và chỉ khi A=B=C
Cách 10 BĐT lượng giác cơ bản
Ta có : [Only registered and activated users can see links]
[Only registered and activated users can see links] ( đẳng thức xảy ra khi A=B)
[Only registered and activated users can see links]
[Only registered and activated users can see links]
( đẳng thức xảy ra khi [Only registered and activated users can see links]
Vậy :[Only registered and activated users can see links]
Đẳng thức xảy ra khi và chỉ khi tam giác ABC đều.
Cách 11 Đánh Giá BĐT
-Tam giác ABC không nhọn, Giả sử góc [Only registered and activated users can see links]
Ta có :[Only registered and activated users can see links]
[Only registered and activated users can see links] (1)
[Only registered and activated users can see links]
[Only registered and activated users can see links] (2)
Cộng (1) và (2) vế theo vế ta có:
[Only registered and activated users can see links]
[Only registered and activated users can see links] (3)
Suy ra [Only registered and activated users can see links]
Nếu A nhọn, thì (1), (2), (3) đều thỏa mãn.
Cách 12 Hàm lồi
Nếu tam giác không nhọn, luôn đúng ! :
Xét hàm số f(x) = cosx trong [Only registered and activated users can see links]
Ta có f'(x) = -sinx , f''(x)=-cosx <0 với [Only registered and activated users can see links]
Do đó hàm f(x) = cosx lồi trên [Only registered and activated users can see links]
Do đó [Only registered and activated users can see links]
[Only registered and activated users can see links]
Đẳng thức xảy ra khi tam giác ABC đều
CÁC CÁCH GIẢI BẤT ĐẢNG THỨC NESBIT
[Only registered and activated users can see links] rac%7Bb%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Bc%7D%7Ba%2Bb%7D %5Cgeq%5Cdfrac%7B3%7D%7B2%7D%5Cforall+a%2Cb%2Cc%3E 0&bg=ffffff&fg=000000&s=0
Cach 1.Cộng thêm 1+1+1 vào vế trái của bất đẳng thức , ta có:
[Only registered and activated users can see links] 7B1%7D%7Ba%2Bb%7D%2B%5Cdfrac%7B1%7D%7Bb%2Bc%7D%2B% 5Cdfrac%7B1%7D%7Bc%2Ba%7D%5Cright%29%5Cgeq+%5Cdfra c%7B9%7D%7B2%7D&bg=ffffff&fg=000000&s=0 ( Đúng)
Cach 2.
Đặt [Only registered and activated users can see links] 5Cdfrac%7Bb%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Bc%7D%7Ba%2B b%7D&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] 5Cdfrac%7Bc%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Ba%7D%7Ba%2B b%7D&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] 5Cdfrac%7Ba%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Bb%7D%7Ba%2B b%7D&bg=ffffff&fg=000000&s=0
Ta có [Only registered and activated users can see links]
[Only registered and activated users can see links] c%7D%2B%5Cdfrac%7Bb%2Bc%7D%7Bc%2Ba%7D%2B%5Cdfrac%7 Bc%2Ba%7D%7Ba%2Bb%7D%5Cge+3&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] c%7D%2B%5Cdfrac%7Bb%2Ba%7D%7Bc%2Ba%7D%2B%5Cdfrac%7 Bc%2Bb%7D%7Ba%2Bb%7D%5Cge+3&bg=ffffff&fg=000000&s=0
Suy ra [Only registered and activated users can see links]
Cach 3.
Không giảm tính tổng quát, giả sử: [Only registered and activated users can see links]
Xét hàm số: [Only registered and activated users can see links] trên [Only registered and activated users can see links] ta có [Only registered and activated users can see links] Do do [Only registered and activated users can see links] la ham so loi tren [Only registered and activated users can see links] theo bat dang thuc Jensen ta co [Only registered and activated users can see links] eq+3f%28%5Cdfrac%7Ba%2Bb%2Bc%7D%7B3%7D%29%3D3f%28% 5Cdfrac%7B1%7D%7B3%7D%29%3D%5Cdfrac%7B3%7D%7B2%7D&bg=ffffff&fg=000000&s=0. Nhu vay bat dang thuc duoc chung minh.[Only registered and activated users can see links]
Cach 4.
Dat [Only registered and activated users can see links] %7Bb%2Bc%7D%2B%5Cdfrac%7Bb%7D%7Ba%2Bc%7D%2B%5Cdfra c%7Bc%7D%7Ba%2Bb%7D&bg=ffffff&fg=000000&s=0
Ta se chung minh:
[Only registered and activated users can see links] %7Ba%2Bb%7D%7B2%7D%2C%5Cdfrac%7Ba%2Bb%7D%7B2%7D%2C c%29&bg=ffffff&fg=000000&s=0
That vay :
[Only registered and activated users can see links] %7D%7B2%7D%2Cc%29%3D%5Cdfrac%7B%28a-b%29%5E2.%28a%2Bb%2Bc%29%7D%7B%282c%2Ba%2Bb%29%28b %2Bc%29%28a%2Bc%29%7D%5Cgeq+0&bg=ffffff&fg=000000&s=0
Vay ta co [Only registered and activated users can see links] %7Ba%2Bb%7D%7B2%7D%2C%5Cdfrac%7Ba%2Bb%7D%7B2%7D%2C c%29&bg=ffffff&fg=000000&s=0
Tiep theo ta chung minh [Only registered and activated users can see links] 5Cdfrac%7Ba%2Bb%7D%7B2%7D%2Cc%29%5Cge%5Cdfrac%7B3% 7D%7B2%7D&bg=ffffff&fg=000000&s=0
That vay Dat [Only registered and activated users can see links]
Bat dang thuc can chung minh tuong duong voi
[Only registered and activated users can see links] frac%7Bc%7D%7B2t%7D%5Cge+%5Cdfrac%7B3%7D%7B2%7D&bg=ffffff&fg=000000&s=0
Hay [Only registered and activated users can see links]
Hien nhien dung.
Cach 5.
Bien doi bat dang thuc ve dang SOS:
[Only registered and activated users can see links]
BDT hien nhien dung vi [Only registered and activated users can see links]
Cach 6.
Bien doi ve dang SS.Gia su [Only registered and activated users can see links]
[Only registered and activated users can see links] c%29%28b%2Bc%29%7D.%28a-b%29%5E2%2B%5C%5C+%5Cdfrac%7Ba%2Bb%2B2c%7D%7B2%28a %2Bb%29%28b%2Bc%29%28c%2Ba%29%7D.%28a-c%29%28b-c%29&bg=ffffff&fg=000000&s=0
Hien nhien dung . Cach 7.
Ap dung bat dang thuc Cauchy-Schwarz ta co :
[Only registered and activated users can see links] D%5Csum%5Cdfrac%7Ba%5E2%7D%7Bab%2Bbc%7D&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] E2%7D%7B2%28ab%2Bbc%2Bca%29%7D&bg=ffffff&fg=000000&s=0
Theo bat dang thuc quen thuoc :
[Only registered and activated users can see links] bc%2Bca%29&bg=ffffff&fg=000000&s=0
Ta co dpcm . Cach 8.
Bat dang thuc tuong duong voi
[Only registered and activated users can see links] 3%2B%5B%5Csum_%7Bsym%7Da%5E2.b%5D%2B3abc%7D%7B%5B% 5Csum_%7Bsym%7Da%5E2.b%5D%2B2abc%7D%5Cge+%5Cdfrac% 7B3%7D%7B2%7D&bg=ffffff&fg=000000&s=0
tuong duong voi :
[Only registered and activated users can see links] Csum_%7Bsym%7Da%5E2.b&bg=ffffff&fg=000000&s=0
Theo AM-GM:
[Only registered and activated users can see links] 5E3%2Bb%5E3%7D%7B3%7D&bg=ffffff&fg=000000&s=0
Cong ve theo ve 6 BDT tuong tu ta co dpcm.
done. Cach 9.
Khong giam tong quat, gia su [Only registered and activated users can see links] khi do [Only registered and activated users can see links] hay [Only registered and activated users can see links] Cdfrac%7B1%7D%7Bc%2Ba%7D%5Cgeq+%5Cdfrac%7B1%7D%7Ba %2Bb%7D&bg=ffffff&fg=000000&s=0 (chu y la [Only registered and activated users can see links]) va boi bat dang thuc hoan vi ta co
[Only registered and activated users can see links] rac%7Bb%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Bc%7D%7Ba%2Bb%7D %5Cgeq+%5Cdfrac%7Bb%7D%7Bb%2Bc%7D%2B%5Cdfrac%7Bc%7 D%7Bc%2Ba%7D%2B%5Cdfrac%7Ba%7D%7Ba%2Bb%7D&bg=ffffff&fg=000000&s=0(1) va [Only registered and activated users can see links] rac%7Bb%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Bc%7D%7Ba%2Bb%7D %5Cgeq+%5Cdfrac%7Bc%7D%7Bb%2Bc%7D%2B%5Cdfrac%7Ba%7 D%7Bc%2Ba%7D%2B%5Cdfrac%7Bb%7D%7Ba%2Bb%7D&bg=ffffff&fg=000000&s=0(2) . Cong theo ve (1) voi (2) ta co dieu phai chung minh. [Only registered and activated users can see links]
Cach 10.
Dat [Only registered and activated users can see links] 2z&bg=ffffff&fg=000000&s=0 suy ra [Only registered and activated users can see links] va bat dang thuc da cho tro thanh [Only registered and activated users can see links] rac%7By%2Bz-x%7D%7Bx%7D%5Cgeq+%5Cdfrac%7B3%7D%7B2%7D&bg=ffffff&fg=000000&s=0 hay [Only registered and activated users can see links] rac%7Bx%7D%7By%7D%5Cgeq+6&bg=ffffff&fg=000000&s=0, bat dang thuc cuoi cung nay dung theo AM-GM(chu y la [Only registered and activated users can see links] Cach 11.
Khong giam tong quat , gia su [Only registered and activated users can see links] Bang bien doi tuong duong ta chung minh duoc bat dang thuc sau [Only registered and activated users can see links](*). Lan luot thay [Only registered and activated users can see links] boi [Only registered and activated users can see links] trong (*) chung ta duoc ba bat dang thuc, cong theo ve ba bat dang thuc nay chung ta duoc dieu phai chung minh.[Only registered and activated users can see links] Cach 12.
Dat [Only registered and activated users can see links] 5Cdfrac%7Bb%7D%7Bc%2Ba%7D%3Dy%2C%5Cdfrac%7Bc%7D%7B a%2Bb%7D%3Dz&bg=ffffff&fg=000000&s=0 thi [Only registered and activated users can see links] va chung ta can chung minh [Only registered and activated users can see links] 2%7D&bg=ffffff&fg=000000&s=0. Gia su nguoc lai [Only registered and activated users can see links] 7D&bg=ffffff&fg=000000&s=0, khi do [Only registered and activated users can see links] frac%7B%28x%2By%2Bz%29%5E2%7D%7B3%7D%2B2%5Cleft%28 %5Cdfrac%7Bx%2By%2Bz%7D%7B3%7D%5Cright%29%5E3%3C1&bg=ffffff&fg=000000&s=0, vo li! Suy ra dieu phai chung minh.[Only registered and activated users can see links] Cach 13.
Chuan hóa cho [Only registered and activated users can see links] Ta có:
[Only registered and activated users can see links] rac%7B9a%28b%2Bc%29%7D%7B4%7D%5Cgeq+3a&bg=ffffff&fg=000000&s=0 tương tu voi các bieu thuc khác sau đó áp dung thêm bdt:
[Only registered and activated users can see links] Bbc%2Bca%29&bg=ffffff&fg=000000&s=0 ta có đpcm . [Only registered and activated users can see links]
Cach 14.
Ta chung minh nhan xet sau
[Only registered and activated users can see links] 5Cdfrac%7B8a-b-c%7D%7B4%28a%2Bb%2Bc%29%7D&bg=ffffff&fg=000000&s=0
That vay,no tuong duong voi:[Only registered and activated users can see links]
hien nhien dung.Cong ve theo ve cac bat dang thuc tuong tu ta co ngay dpcm.
done.
Cach 15.
Gia su [Only registered and activated users can see links] ra:
[Only registered and activated users can see links] Cdfrac%7B1%7D%7Bc%2Ba%7D%5Cge+%5Cdfrac%7B1%7D%7Ba% 2Bb%7D&bg=ffffff&fg=000000&s=0
Theo bat dang thuc Chebyshev,ta co :
[Only registered and activated users can see links] rac%7Bb%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Bc%7D%7Ba%2Bb%7D +%5Cge+&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] c%29%28%5Cdfrac%7B1%7D%7Ba%2Bb%7D%2B%5Cdfrac%7B1%7 D%7Bb%2Bc%7D%2B%5Cdfrac%7B1%7D%7Bc%2Ba%7D%29+%3D+&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] 29%2B%28b%2Bc%29%2B%28c%2Ba%29%5D%5B%5Cdfrac%7B1%7 D%7Ba%2Bb%7D%2B%5Cdfrac%7B1%7D%7Bb%2Bc%7D%2B%5Cdfr ac%7B1%7D%7Bc%2Ba%7D%5D%5Cge%5Cdfrac%7B9%7D%7B6%7D +&bg=ffffff&fg=000000&s=0 ( theo AM-GM)
ta co dieu can phai chung minh.
Cach 16.
Theo Schwarz ta co :
[Only registered and activated users can see links] %7Bb%2Bc%7D%3D%5Csum_%7Bcyclic%7D%5Cdfrac%7Ba%5E4% 7D%7Ba%5E3%28b%2Bc%29%7D&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] Bc%5E2%29%5E2%7D%7B%5Csum_%7Bsym%7Da%5E3.b%7D%5Cge +%5Cdfrac%7B3%7D%7B2%7D&bg=ffffff&fg=000000&s=0
Ta can chung minh :[Only registered and activated users can see links] Cge+3%5Csum_%7Bsym%7Da%5E3.b&bg=ffffff&fg=000000&s=0
Day la 1 bat dang thuc rat noi tieng cua Vasile Cirtoaje.
Cach 17.
[Only registered and activated users can see links] %7D+%2B+%5Cdfrac%7Ba%5E%7B2%7D%7D%7Bb%2Bc%7D+%2B+% 5Cdfrac%7Bb%2Bc%7D%7B4%7D+%5Cgeq+2a&bg=ffffff&fg=000000&s=0
suy ra
[Only registered and activated users can see links] %2Bc%7D+%2B+%5Cdfrac%7Bb%2Bc%7D%7B4%7D++%5Cgeq+2a&bg=ffffff&fg=000000&s=0
tuong tu
[Only registered and activated users can see links] %2Bc%7D+%2B+%5Cdfrac%7Ba%2Bc%7D%7B4%7D++%5Cgeq+2b&bg=ffffff&fg=000000&s=0
va
[Only registered and activated users can see links] a%2Bb%7D+%2B+%5Cdfrac%7Bb%2Ba%7D%7B4%7D++%5Cgeq+2c&bg=ffffff&fg=000000&s=0
cong vao :
[Only registered and activated users can see links] %7Bb%2Bc%7D+%2B+%5Cdfrac%7Bb%7D%7Bc%2Ba%7D+%2B+%5C dfrac%7Bc%7D%7Bb%2Ba%7D+%29+%5Cgeq++%5Cdfrac%7B3%7 D%7B2%7D%28a%2Bb%2Bc%29&bg=ffffff&fg=000000&s=0
Cach 18
Dat [Only registered and activated users can see links] %3D%5Cdfrac%7Bb%7D%7Bc%2Ba%7D%2Cz%3D%5Cdfrac%7Bc%7 D%7Ba%2Bb%7D&bg=ffffff&fg=000000&s=0 va xet ham so [Only registered and activated users can see links] %7D%28t%3E0%29&bg=ffffff&fg=000000&s=0. Ta co [Only registered and activated users can see links] 8t%2B1%29%5E2%7D%3E0%5Cforall+t%3E0&bg=ffffff&fg=000000&s=0 va [Only registered and activated users can see links] t%5Cin+%280%2C%5Cinfty%29&bg=ffffff&fg=000000&s=0. Theo bat dang thuc Jensen ta co [Only registered and activated users can see links] Cdfrac%7B1%7D%7B3%7D%3D%5Cdfrac%7Bf%28x%29%2Bf%28y %29%2Bf%28z%29%7D%7B3%7D%5Cleq+f%5Cleft%28%5Cdfrac %7Bx%2By%2Bz%7D%7B3%7D%5Cright%29&bg=ffffff&fg=000000&s=0. Nhung vi [Only registered and activated users can see links] la tang ngat tren [Only registered and activated users can see links] nen [Only registered and activated users can see links] rac%7Bx%2By%2Bz%7D%7B3%7D&bg=ffffff&fg=000000&s=0. Suy ra dieu phai chung minh.[Only registered and activated users can see links]
Cach 19.
Se la du neu ta chung minh duoc [Only registered and activated users can see links] 5Cdfrac%7B3a%5E%7B3%2F2%7D%7D%7B2%28a%5E%7B3%2F2%7 D%2Bb%5E%7B3%2F2%7D%2Bc%5E%7B3%2F2%7D%29%7D&bg=ffffff&fg=000000&s=0 hay [Only registered and activated users can see links] %7D%2Bc%5E%7B3%2F2%7D%29%5Cgeq+3a%5E%7B1%2F2%7D%28 b%2Bc%29&bg=ffffff&fg=000000&s=0(*).
Theo AM-GM ta co
[Only registered and activated users can see links] 2Bb%5E%7B3%2F2%7D%5Cgeq+3a%5E%7B1%2F2%7Db&bg=ffffff&fg=000000&s=0 va
[Only registered and activated users can see links] 2Bc%5E%7B3%2F2%7D%5Cgeq+3a%5E%7B1%2F2%7Dc&bg=ffffff&fg=000000&s=0 .
Cong theo ve hai bat dang thuc cuoi nay ta duoc (*).[Only registered and activated users can see links]
Cách 20
[Only registered and activated users can see links] Bb%2Bc%7D%3D%5Cdfrac%7B1%7D%7B2%7D%5Csum_%7Bcycl%7 D%5Cdfrac%7B2a%7D%7Bb%2Bc%7D&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] Bcycl%7D%5Cleft%28%5Cdfrac%7B%28c%2Ba%29%2B%28a%2B b%29%7D%7Bb%2Bc%7D-1%5Cright%29&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links]
Cach 21.
[Only registered and activated users can see links] c%7Ba%7D%7Bb%2Bc%7D-%5Cdfrac%7B1%7D%7B2%7D%5Cright%29&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] 28b%2Bc%29%28c%2Ba%29%7D%5Csum_%7Bcycl%7D+%282a%2B b%2Bc%29%28a-b%29%28a-c%29+&bg=ffffff&fg=000000&s=0
CHO BẤT ĐẢNG THỨC ĐƠN GIẢN
Chứng minh rằng [Only registered and activated users can see links]
trong đó A, B, C là ba góc của một tam giác bất kì .
( Theo thứ tự chương trình học Phổ thông )
Cách 1 Dùng tỉ số Diện Tích
Kẻ các đường cao AD, BE, CF
Đặt [Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links] [Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
Tương tự
[Only registered and activated users can see links]
[Only registered and activated users can see links]
Cộng (1), (2), (3) ta có
[Only registered and activated users can see links]
[Only registered and activated users can see links](đpcm )
Đẳng thức xảy ra khi và chỉ khi tam giác ABC đều.
Cách 2: Vận dụng bất đẳng thức :Erdos-Mordell
Cho tam giác ABC. M là một điểm bất kì nằm trong tam giác .
Đặt [Only registered and activated users can see links]
và [Only registered and activated users can see links] lần lượt là khoảng cách từ M đến BC, CA, AB tương ứng.
Khi đó ta có bất đẳng thức [Only registered and activated users can see links]
Vận dụng giải bài trên:
Gọi O , R là tâm và bán kính đường tròn ngoại tiếp tam giác ABC.
Gọi M, N, P lần lượt là trung điểm của cạnh AB, BC, CA..
Ta dễ dàng nhận thấy [Only registered and activated users can see links]
Do đó :[Only registered and activated users can see links]
Tương tự [Only registered and activated users can see links]
Do đó [Only registered and activated users can see links]
[Only registered and activated users can see links] ( đpcm).(Erdos-Mordell)
Đẳng thức xảy ra khi và chỉ khi tam giác ABC đều.
Cách 3 Sử dụng BĐT Trêbưsep.
Gọi a, b, c là ba cạnh tam giác, sử dụng công thức hình chiếu ta có:
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
Cộng ba biểu thức trên ta có:
[Only registered and activated users can see links]
Không mất tính tổng quát giả sử: [Only registered and activated users can see links] ta có:
[Only registered and activated users can see links]
Do đó :[Only registered and activated users can see links]
[Only registered and activated users can see links] ( Trêbưsep)
[Only registered and activated users can see links] (đpcm)
Đẳng thức xảy ra khi tam giác ABC đều.
Cách 4 Phuong pháp vectơ.
Gọi I và r lần lượt là tâm và bán kính đường tròn nội tiếp tam giác ABC,
và M, N, P lần lượt là tiếp điểm của đường tròn đó
với các cạnh AB, AC, BC ,ta có
[Only registered and activated users can see links]
[Only registered and activated users can see links] (*)
Ta nhận thấy
[Only registered and activated users can see links]
( Vì [Only registered and activated users can see links] và góc A bù nhau)
Tương tự :[Only registered and activated users can see links] [Only registered and activated users can see links]
Vậy từ (*) suy ra [Only registered and activated users can see links] (dpcm)
Cách 5: Phuong pháp vectơ.
Lấy A, B, C lần lượt là ba gốc của ba véctơ đơn vị sau
[Only registered and activated users can see links] [Only registered and activated users can see links]
[Only registered and activated users can see links]
Ta có :[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links] [Only registered and activated users can see links]
Cách 6: Quan hệ bất đẳng thức Schur.
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]( Schur)
Cách 7 Sử dụng tam thức bậc hai.
Xét [Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links]
Đặt [Only registered and activated users can see links]
Xét tam thức [Only registered and activated users can see links]
Có [Only registered and activated users can see links]
và hệ số [Only registered and activated users can see links]ên [Only registered and activated users can see links] với mọi x
Hay [Only registered and activated users can see links]
Cách 8: Sử dụng hàm số.
Ta có [Only registered and activated users can see links]
[Only registered and activated users can see links]
Đặt [Only registered and activated users can see links]
điều kiện [Only registered and activated users can see links]ét hàm số [Only registered and activated users can see links]
Lập bảng xét dấu ta có [Only registered and activated users can see links]
Cách 9 Tổng bình phương.
Xét [Only registered and activated users can see links]
[Only registered and activated users can see links]
[Only registered and activated users can see links] (Đúng)
Đẳng thức xảy ra khi và chỉ khi A=B=C
Cách 10 BĐT lượng giác cơ bản
Ta có : [Only registered and activated users can see links]
[Only registered and activated users can see links] ( đẳng thức xảy ra khi A=B)
[Only registered and activated users can see links]
[Only registered and activated users can see links]
( đẳng thức xảy ra khi [Only registered and activated users can see links]
Vậy :[Only registered and activated users can see links]
Đẳng thức xảy ra khi và chỉ khi tam giác ABC đều.
Cách 11 Đánh Giá BĐT
-Tam giác ABC không nhọn, Giả sử góc [Only registered and activated users can see links]
Ta có :[Only registered and activated users can see links]
[Only registered and activated users can see links] (1)
[Only registered and activated users can see links]
[Only registered and activated users can see links] (2)
Cộng (1) và (2) vế theo vế ta có:
[Only registered and activated users can see links]
[Only registered and activated users can see links] (3)
Suy ra [Only registered and activated users can see links]
Nếu A nhọn, thì (1), (2), (3) đều thỏa mãn.
Cách 12 Hàm lồi
Nếu tam giác không nhọn, luôn đúng ! :
Xét hàm số f(x) = cosx trong [Only registered and activated users can see links]
Ta có f'(x) = -sinx , f''(x)=-cosx <0 với [Only registered and activated users can see links]
Do đó hàm f(x) = cosx lồi trên [Only registered and activated users can see links]
Do đó [Only registered and activated users can see links]
[Only registered and activated users can see links]
Đẳng thức xảy ra khi tam giác ABC đều
CÁC CÁCH GIẢI BẤT ĐẢNG THỨC NESBIT
[Only registered and activated users can see links] rac%7Bb%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Bc%7D%7Ba%2Bb%7D %5Cgeq%5Cdfrac%7B3%7D%7B2%7D%5Cforall+a%2Cb%2Cc%3E 0&bg=ffffff&fg=000000&s=0
Cach 1.Cộng thêm 1+1+1 vào vế trái của bất đẳng thức , ta có:
[Only registered and activated users can see links] 7B1%7D%7Ba%2Bb%7D%2B%5Cdfrac%7B1%7D%7Bb%2Bc%7D%2B% 5Cdfrac%7B1%7D%7Bc%2Ba%7D%5Cright%29%5Cgeq+%5Cdfra c%7B9%7D%7B2%7D&bg=ffffff&fg=000000&s=0 ( Đúng)
Cach 2.
Đặt [Only registered and activated users can see links] 5Cdfrac%7Bb%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Bc%7D%7Ba%2B b%7D&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] 5Cdfrac%7Bc%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Ba%7D%7Ba%2B b%7D&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] 5Cdfrac%7Ba%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Bb%7D%7Ba%2B b%7D&bg=ffffff&fg=000000&s=0
Ta có [Only registered and activated users can see links]
[Only registered and activated users can see links] c%7D%2B%5Cdfrac%7Bb%2Bc%7D%7Bc%2Ba%7D%2B%5Cdfrac%7 Bc%2Ba%7D%7Ba%2Bb%7D%5Cge+3&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] c%7D%2B%5Cdfrac%7Bb%2Ba%7D%7Bc%2Ba%7D%2B%5Cdfrac%7 Bc%2Bb%7D%7Ba%2Bb%7D%5Cge+3&bg=ffffff&fg=000000&s=0
Suy ra [Only registered and activated users can see links]
Cach 3.
Không giảm tính tổng quát, giả sử: [Only registered and activated users can see links]
Xét hàm số: [Only registered and activated users can see links] trên [Only registered and activated users can see links] ta có [Only registered and activated users can see links] Do do [Only registered and activated users can see links] la ham so loi tren [Only registered and activated users can see links] theo bat dang thuc Jensen ta co [Only registered and activated users can see links] eq+3f%28%5Cdfrac%7Ba%2Bb%2Bc%7D%7B3%7D%29%3D3f%28% 5Cdfrac%7B1%7D%7B3%7D%29%3D%5Cdfrac%7B3%7D%7B2%7D&bg=ffffff&fg=000000&s=0. Nhu vay bat dang thuc duoc chung minh.[Only registered and activated users can see links]
Cach 4.
Dat [Only registered and activated users can see links] %7Bb%2Bc%7D%2B%5Cdfrac%7Bb%7D%7Ba%2Bc%7D%2B%5Cdfra c%7Bc%7D%7Ba%2Bb%7D&bg=ffffff&fg=000000&s=0
Ta se chung minh:
[Only registered and activated users can see links] %7Ba%2Bb%7D%7B2%7D%2C%5Cdfrac%7Ba%2Bb%7D%7B2%7D%2C c%29&bg=ffffff&fg=000000&s=0
That vay :
[Only registered and activated users can see links] %7D%7B2%7D%2Cc%29%3D%5Cdfrac%7B%28a-b%29%5E2.%28a%2Bb%2Bc%29%7D%7B%282c%2Ba%2Bb%29%28b %2Bc%29%28a%2Bc%29%7D%5Cgeq+0&bg=ffffff&fg=000000&s=0
Vay ta co [Only registered and activated users can see links] %7Ba%2Bb%7D%7B2%7D%2C%5Cdfrac%7Ba%2Bb%7D%7B2%7D%2C c%29&bg=ffffff&fg=000000&s=0
Tiep theo ta chung minh [Only registered and activated users can see links] 5Cdfrac%7Ba%2Bb%7D%7B2%7D%2Cc%29%5Cge%5Cdfrac%7B3% 7D%7B2%7D&bg=ffffff&fg=000000&s=0
That vay Dat [Only registered and activated users can see links]
Bat dang thuc can chung minh tuong duong voi
[Only registered and activated users can see links] frac%7Bc%7D%7B2t%7D%5Cge+%5Cdfrac%7B3%7D%7B2%7D&bg=ffffff&fg=000000&s=0
Hay [Only registered and activated users can see links]
Hien nhien dung.
Cach 5.
Bien doi bat dang thuc ve dang SOS:
[Only registered and activated users can see links]
BDT hien nhien dung vi [Only registered and activated users can see links]
Cach 6.
Bien doi ve dang SS.Gia su [Only registered and activated users can see links]
[Only registered and activated users can see links] c%29%28b%2Bc%29%7D.%28a-b%29%5E2%2B%5C%5C+%5Cdfrac%7Ba%2Bb%2B2c%7D%7B2%28a %2Bb%29%28b%2Bc%29%28c%2Ba%29%7D.%28a-c%29%28b-c%29&bg=ffffff&fg=000000&s=0
Hien nhien dung . Cach 7.
Ap dung bat dang thuc Cauchy-Schwarz ta co :
[Only registered and activated users can see links] D%5Csum%5Cdfrac%7Ba%5E2%7D%7Bab%2Bbc%7D&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] E2%7D%7B2%28ab%2Bbc%2Bca%29%7D&bg=ffffff&fg=000000&s=0
Theo bat dang thuc quen thuoc :
[Only registered and activated users can see links] bc%2Bca%29&bg=ffffff&fg=000000&s=0
Ta co dpcm . Cach 8.
Bat dang thuc tuong duong voi
[Only registered and activated users can see links] 3%2B%5B%5Csum_%7Bsym%7Da%5E2.b%5D%2B3abc%7D%7B%5B% 5Csum_%7Bsym%7Da%5E2.b%5D%2B2abc%7D%5Cge+%5Cdfrac% 7B3%7D%7B2%7D&bg=ffffff&fg=000000&s=0
tuong duong voi :
[Only registered and activated users can see links] Csum_%7Bsym%7Da%5E2.b&bg=ffffff&fg=000000&s=0
Theo AM-GM:
[Only registered and activated users can see links] 5E3%2Bb%5E3%7D%7B3%7D&bg=ffffff&fg=000000&s=0
Cong ve theo ve 6 BDT tuong tu ta co dpcm.
done. Cach 9.
Khong giam tong quat, gia su [Only registered and activated users can see links] khi do [Only registered and activated users can see links] hay [Only registered and activated users can see links] Cdfrac%7B1%7D%7Bc%2Ba%7D%5Cgeq+%5Cdfrac%7B1%7D%7Ba %2Bb%7D&bg=ffffff&fg=000000&s=0 (chu y la [Only registered and activated users can see links]) va boi bat dang thuc hoan vi ta co
[Only registered and activated users can see links] rac%7Bb%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Bc%7D%7Ba%2Bb%7D %5Cgeq+%5Cdfrac%7Bb%7D%7Bb%2Bc%7D%2B%5Cdfrac%7Bc%7 D%7Bc%2Ba%7D%2B%5Cdfrac%7Ba%7D%7Ba%2Bb%7D&bg=ffffff&fg=000000&s=0(1) va [Only registered and activated users can see links] rac%7Bb%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Bc%7D%7Ba%2Bb%7D %5Cgeq+%5Cdfrac%7Bc%7D%7Bb%2Bc%7D%2B%5Cdfrac%7Ba%7 D%7Bc%2Ba%7D%2B%5Cdfrac%7Bb%7D%7Ba%2Bb%7D&bg=ffffff&fg=000000&s=0(2) . Cong theo ve (1) voi (2) ta co dieu phai chung minh. [Only registered and activated users can see links]
Cach 10.
Dat [Only registered and activated users can see links] 2z&bg=ffffff&fg=000000&s=0 suy ra [Only registered and activated users can see links] va bat dang thuc da cho tro thanh [Only registered and activated users can see links] rac%7By%2Bz-x%7D%7Bx%7D%5Cgeq+%5Cdfrac%7B3%7D%7B2%7D&bg=ffffff&fg=000000&s=0 hay [Only registered and activated users can see links] rac%7Bx%7D%7By%7D%5Cgeq+6&bg=ffffff&fg=000000&s=0, bat dang thuc cuoi cung nay dung theo AM-GM(chu y la [Only registered and activated users can see links] Cach 11.
Khong giam tong quat , gia su [Only registered and activated users can see links] Bang bien doi tuong duong ta chung minh duoc bat dang thuc sau [Only registered and activated users can see links](*). Lan luot thay [Only registered and activated users can see links] boi [Only registered and activated users can see links] trong (*) chung ta duoc ba bat dang thuc, cong theo ve ba bat dang thuc nay chung ta duoc dieu phai chung minh.[Only registered and activated users can see links] Cach 12.
Dat [Only registered and activated users can see links] 5Cdfrac%7Bb%7D%7Bc%2Ba%7D%3Dy%2C%5Cdfrac%7Bc%7D%7B a%2Bb%7D%3Dz&bg=ffffff&fg=000000&s=0 thi [Only registered and activated users can see links] va chung ta can chung minh [Only registered and activated users can see links] 2%7D&bg=ffffff&fg=000000&s=0. Gia su nguoc lai [Only registered and activated users can see links] 7D&bg=ffffff&fg=000000&s=0, khi do [Only registered and activated users can see links] frac%7B%28x%2By%2Bz%29%5E2%7D%7B3%7D%2B2%5Cleft%28 %5Cdfrac%7Bx%2By%2Bz%7D%7B3%7D%5Cright%29%5E3%3C1&bg=ffffff&fg=000000&s=0, vo li! Suy ra dieu phai chung minh.[Only registered and activated users can see links] Cach 13.
Chuan hóa cho [Only registered and activated users can see links] Ta có:
[Only registered and activated users can see links] rac%7B9a%28b%2Bc%29%7D%7B4%7D%5Cgeq+3a&bg=ffffff&fg=000000&s=0 tương tu voi các bieu thuc khác sau đó áp dung thêm bdt:
[Only registered and activated users can see links] Bbc%2Bca%29&bg=ffffff&fg=000000&s=0 ta có đpcm . [Only registered and activated users can see links]
Cach 14.
Ta chung minh nhan xet sau
[Only registered and activated users can see links] 5Cdfrac%7B8a-b-c%7D%7B4%28a%2Bb%2Bc%29%7D&bg=ffffff&fg=000000&s=0
That vay,no tuong duong voi:[Only registered and activated users can see links]
hien nhien dung.Cong ve theo ve cac bat dang thuc tuong tu ta co ngay dpcm.
done.
Cach 15.
Gia su [Only registered and activated users can see links] ra:
[Only registered and activated users can see links] Cdfrac%7B1%7D%7Bc%2Ba%7D%5Cge+%5Cdfrac%7B1%7D%7Ba% 2Bb%7D&bg=ffffff&fg=000000&s=0
Theo bat dang thuc Chebyshev,ta co :
[Only registered and activated users can see links] rac%7Bb%7D%7Bc%2Ba%7D%2B%5Cdfrac%7Bc%7D%7Ba%2Bb%7D +%5Cge+&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] c%29%28%5Cdfrac%7B1%7D%7Ba%2Bb%7D%2B%5Cdfrac%7B1%7 D%7Bb%2Bc%7D%2B%5Cdfrac%7B1%7D%7Bc%2Ba%7D%29+%3D+&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] 29%2B%28b%2Bc%29%2B%28c%2Ba%29%5D%5B%5Cdfrac%7B1%7 D%7Ba%2Bb%7D%2B%5Cdfrac%7B1%7D%7Bb%2Bc%7D%2B%5Cdfr ac%7B1%7D%7Bc%2Ba%7D%5D%5Cge%5Cdfrac%7B9%7D%7B6%7D +&bg=ffffff&fg=000000&s=0 ( theo AM-GM)
ta co dieu can phai chung minh.
Cach 16.
Theo Schwarz ta co :
[Only registered and activated users can see links] %7Bb%2Bc%7D%3D%5Csum_%7Bcyclic%7D%5Cdfrac%7Ba%5E4% 7D%7Ba%5E3%28b%2Bc%29%7D&bg=ffffff&fg=000000&s=0
[Only registered and activated users can see links] Bc%5E2%29%5E2%7D%7B%5Csum_%7Bsym%7Da%5E3.b%7D%5Cge +%5Cdfrac%7B3%7D%7B2%7D&bg=ffffff&fg=000000&s=0
Ta can chung minh :[Only registered and activated users can see links] Cge+3%5Csum_%7Bsym%7Da%5E3.b&bg=ffffff&fg=000000&s=0
Day la 1 bat dang thuc rat noi tieng cua Vasile Cirtoaje.
Cach 17.
[Only registered and activated users can see links] %7D+%2B+%5Cdfrac%7Ba%5E%7B2%7D%7D%7Bb%2Bc%7D+%2B+% 5Cdfrac%7Bb%2Bc%7D%7B4%7D+%5Cgeq+2a&bg=ffffff&fg=000000&s=0
suy ra
[Only registered and activated users can see links] %2Bc%7D+%2B+%5Cdfrac%7Bb%2Bc%7D%7B4%7D++%5Cgeq+2a&bg=ffffff&fg=000000&s=0
tuong tu
[Only registered and activated users can see links] %2Bc%7D+%2B+%5Cdfrac%7Ba%2Bc%7D%7B4%7D++%5Cgeq+2b&bg=ffffff&fg=000000&s=0
va
[Only registered and activated users can see links] a%2Bb%7D+%2B+%5Cdfrac%7Bb%2Ba%7D%7B4%7D++%5Cgeq+2c&bg=ffffff&fg=000000&s=0
cong vao :
[Only registered and activated users can see links] %7Bb%2Bc%7D+%2B+%5Cdfrac%7Bb%7D%7Bc%2Ba%7D+%2B+%5C dfrac%7Bc%7D%7Bb%2Ba%7D+%29+%5Cgeq++%5Cdfrac%7B3%7 D%7B2%7D%28a%2Bb%2Bc%29&bg=ffffff&fg=000000&s=0
Cach 18
Dat [Only registered and activated users can see links] %3D%5Cdfrac%7Bb%7D%7Bc%2Ba%7D%2Cz%3D%5Cdfrac%7Bc%7 D%7Ba%2Bb%7D&bg=ffffff&fg=000000&s=0 va xet ham so [Only registered and activated users can see links] %7D%28t%3E0%29&bg=ffffff&fg=000000&s=0. Ta co [Only registered and activated users can see links] 8t%2B1%29%5E2%7D%3E0%5Cforall+t%3E0&bg=ffffff&fg=000000&s=0 va [Only registered and activated users can see links] t%5Cin+%280%2C%5Cinfty%29&bg=ffffff&fg=000000&s=0. Theo bat dang thuc Jensen ta co [Only registered and activated users can see links] Cdfrac%7B1%7D%7B3%7D%3D%5Cdfrac%7Bf%28x%29%2Bf%28y %29%2Bf%28z%29%7D%7B3%7D%5Cleq+f%5Cleft%28%5Cdfrac %7Bx%2By%2Bz%7D%7B3%7D%5Cright%29&bg=ffffff&fg=000000&s=0. Nhung vi [Only registered and activated users can see links] la tang ngat tren [Only registered and activated users can see links] nen [Only registered and activated users can see links] rac%7Bx%2By%2Bz%7D%7B3%7D&bg=ffffff&fg=000000&s=0. Suy ra dieu phai chung minh.[Only registered and activated users can see links]
Cach 19.
Se la du neu ta chung minh duoc [Only registered and activated users can see links] 5Cdfrac%7B3a%5E%7B3%2F2%7D%7D%7B2%28a%5E%7B3%2F2%7 D%2Bb%5E%7B3%2F2%7D%2Bc%5E%7B3%2F2%7D%29%7D&bg=ffffff&fg=000000&s=0 hay [Only registered and activated users can see links] %7D%2Bc%5E%7B3%2F2%7D%29%5Cgeq+3a%5E%7B1%2F2%7D%28 b%2Bc%29&bg=ffffff&fg=000000&s=0(*).
Theo AM-GM ta co
[Only registered and activated users can see links] 2Bb%5E%7B3%2F2%7D%5Cgeq+3a%5E%7B1%2F2%7Db&bg=ffffff&fg=000000&s=0 va
[Only registered and activated users can see links] 2Bc%5E%7B3%2F2%7D%5Cgeq+3a%5E%7B1%2F2%7Dc&bg=ffffff&fg=000000&s=0 .
Cong theo ve hai bat dang thuc cuoi nay ta duoc (*).[Only registered and activated users can see links]
Cách 20
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Cach 21.
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