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In the triangle ABC, the lengths of three sides opposite to angles A, B, and C are denoted by a, b and c respectively. If 
ab cos C + bc cos A + ca cos B = c^2, 
what is the area of ABC? 

 Sep 11, 2022
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The cosine rule says that

 \(\displaystyle \cos C= \frac{a^{2}+b^{2}-c^{2}}{2ab}.\)

Make that substitution and, in the same way, substitute for cos B and cos A.

Tidy up and you find that

\(\displaystyle a^{2}+b^{2}=c^{2},\)

so ABC is a right-angled triangle with hypotenuse c.

 Sep 13, 2022

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