Question

just a small challenge for the smart ones given: cos^2(B)+cos^2(A)+cow^2(C)=1 . Prove the triangle is isosceles

Answer 1

To establish that the triangle is isosceles you need to show that two angles are congruent, and that the third supplements the sum of the congruent ones. (In terms of the picture, if we assume \(B=C\), we need to show \(A=180-2B\).)

Answer 2

Alternatively we can try to establish that two sides are congruent, and the third is not (i.e. \(b=c\not=a\)).

Answer 3

I'm tempted to use the law of cosines here, which rewrites the identity as
\[\begin{align*}
1&=\cos^2A+\cos^2B+\cos^2C\\\\
&=\left(\frac{b^2+c^2-a^2}{2bc}\right)^2+\left(\frac{a^2+c^2-b^2}{2ac}\right)^2+\left(\frac{a^2+b^2-c^2}{2ab}\right)^2
\end{align*}\]
but I'm not sure if this will lead to something fruitful.