If we circumscribe a triangle about a circle and then using the points of the tangency we inscribe a triangle in the circle, does these two triangles have to be similar? I think that their length sides doesn't have to be proportional but I don't know how to prove it.
2026-05-16 22:51:29.1778971889
Circumscribed and Inscribed Circle
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The smaller one is called the intouch or contact triangle of the larger triangle. According to Wolfram MathWorld, $$a'=(-a+b+c)\cos \frac{A}{2}.$$ It should be easy to find a counterexample to similarity from that.