1) Let I be the incenter of acute ΔABC and A' the circumcenter of ΔIBC.

Prove that A, I, and A' are collinear.

2) Let B' and C' be the circumcenters of triangles ICA and IAB respectively.

Prove that the sum of the vectors

**IA'**,

**IB'**, and

**IC'**is the zero vector

**0**

if and only if ΔABC is equilateral.