2016 USAJMO Problems/Problem 1

Problem

The isosceles triangle $\triangle ABC$ , with $AB=AC$ , is inscribed in the circle $\omega$ . Let $P$ be a variable point on the arc $\stackrel{\frown}{BC}$ that does not contain $A$ , and let $I_B$ and $I_C$ denote the incenters of triangles $\triangle ABP$ and $\triangle ACP$ , respectively.

Prove that as $P$ varies, the circumcircle of triangle $\triangle PI_BI_C$ passes through a fixed point.

Solution

We will use complex numbers, with the circumcircle of $\triangle ABC$ as the unit circle. Let $A=1, B=w^2,C=\frac{1}{w^2}, P=p^2,$ such that $I_B=w+wp-p, I_C=p-\frac{p}{w}+\frac{1}{w}.$ We claim that the circumcircle of $\triangle PI_BI_C$ passes through $M=-1.$ This is true iff $k=\frac{(I_B-M)(I_C-P)}{(I_B-P)(I_C-M)}=\frac{(w+wp-p+1)(p+\frac{1}{w}-\frac{p}{w}-p^2)}{(w+wp-p-p^2)(p+\frac{1}{w}-\frac{p}{w}+1)}=\frac{(wp+1)(1-p)}{(p+1)(w-p)}$ is real. This is true iff $k=\overline{k}.$ We can compute $\overline{k}=\frac{(1+\frac{1}{wp})(1-\frac{1}{p})}{(\frac{1}{p}+1)(\frac{1}{w}-\frac{1}{p})}=k,$ so we are done. $\blacksquare$

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2016 USAJMO Problems/Problem 1

Problem

Solution