Problem

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

Tags: geometry, Inscribed circle, excircle, collinear, concurrent



In $\Delta ABC$ points $A_1$, $B_1$, and $C_1$ are the tangential points of the excircles of $ABC$ with its sides. a) Prove that $AA_1$, $BB_1$, and $CC_1$ intersect in one point $N$. b) If $AC+BC=3AB$, prove that the center of the inscribed circle of $ABC$, its tangential point with $AB$, and the point $N$ are collinear.