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# Trigonometry

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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