Cho ▲ABC nhọn <A bằng 60o Chứng minh BC2=AB2 + AC2 -AB.AC
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

Hạ đường cao AH của △ABC
⇒AH⊥BC
Vì △ABC nhọn
⇒Điểm H nằm giữa 2 điểm B và C
Diện tích △ABC là: SABC=\(\dfrac{1}{2}\).BC.AH(1)
Áp dụng hệ thức giữa cạnh và góc vào △AHB(H=900 ),ta có:
AH=AB.\(\sin B\)(2)
Từ (1) và (2)⇒SABC=BC.AB.\(\sin B\)(đpcm)
Bài 1:
Kẻ đường cao $AH$ của tam giác $ABC$. Ta có:
\(\sin A=\frac{BH}{AB}\)
Mà \(\frac{1}{2}BH.AC=S_{ABC}\Rightarrow BH=\frac{2S_{ABC}}{AC}\)
\(\Rightarrow \sin A=\frac{2S_{ABC}}{AB.AC}\)
\(\Rightarrow \frac{BC}{\sin A}=\frac{AB.AC.BC}{2_{ABC}}\)
Hoàn toàn tương tự, kẻ đường cao từ đỉnh $B,C$ , cuối cùng ta có:
\(\frac{BC}{\sin A}=\frac{AC}{\sin B}=\frac{AB}{\sin C}=\frac{AB.BC.AC}{2S_{ABC}}\)
Vậy ta có đpcm.



sin A/2=sinBAD
xét tam giác AKB vuông tại K,có:
sinBAD=BK/AB (1)
xét tam giác BKD vuông tại K,có
BK<=BD thay vào (1):
sinBAD<=BD/AB(2)
lại có:BD/CD=AB/AC
=>BD/(BD+CD)=AB/(AB+AC)
=>BD/BC=AB/(AB+AC)
=>BD=(AB*BC)/(AB+AC) thay vào (2)
sinBAD<=[(AB*BC)/(AB+AC)]/AB
= BC/(AB + AC)
=>ĐPC

Kẻ AD⊥BC tại D, BE⊥AC tại E
Xét ΔBAE vuông tại E có \(\sin BAE=\frac{BE}{AB}\)
=>\(BE=AB\cdot\sin BAC\)
Xét ΔABC có BE là đường cao
nên \(S_{ABC}=\frac12\cdot BE\cdot AC=\frac12\cdot AB\cdot\sin BAC\cdot AC=\frac12\cdot b\cdot c\cdot\sin A\left(1\right)\)
Xét ΔBEC vuông tại E có \(\sin C=\frac{BE}{BC}\)
=>\(BE=BC\cdot\sin C\)
Xét ΔABC có BE là đường cao
nên \(S_{ABC}=\frac12\cdot BE\cdot AC=\frac12\cdot BC\cdot\sin C\cdot AC=\frac12\cdot CA\cdot CB\cdot\sin C=\frac12\cdot a\cdot b\cdot\sin C\) (2)
Xét ΔADB vuông tại D có \(\sin ABD=\frac{AD}{AB}\)
=>\(AD=AB\cdot\sin ABD=AD\cdot\sin ABC\)
Xét ΔABC có AD là đường cao
nên \(S_{ABC}=\frac12\cdot AD\cdot BC=\frac12\cdot AB\cdot\sin ABC\cdot BC=\frac12\cdot BA\cdot BC\cdot\sin ABC=\frac12\cdot c\cdot a\cdot\sin B\) (3)
Từ (1),(2),(3) suy ra \(S_{ABC}=\frac12\cdot ab\cdot\sin C=\frac12\cdot ac\cdot\sin B=\frac12\cdot bc\cdot\sin A\)

Xét tứ giác BDEC có:
∠(BEC) = ∠(BDC) = 900900
Mà 2 góc này cùng nhìn cạnh BC
⇒ Tứ giác BDEC là tứ giác nội tiếp
Xét ΔABC có \(cosA=\dfrac{AB^2+AC^2-BC^2}{2\cdot AB\cdot AC}=\dfrac{1}{2}\)
\(\Leftrightarrow AB^2+AC^2-BC^2=AB\cdot AC\)
hay \(BC^2=BA^2+AC^2-AB\cdot AC\)