Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.8 AIME Prove it Belarus

Let $ABC$ be an isosceles triangle with the base $BC$ . Points $X$ , $Y$ and $Z$ are chosen on the sides $BC$ , $AC$ and $AB$ , respectively, such that $\triangle ABC \sim \triangle YXZ$ . Let $W$ be the reflection of $X$ with respect to the midpoint of the segment $BC$ .
Prove that the points $X$ , $Y$ , $Z$ and $W$ are cocyclic.

Solution

Denote by $O$ the center of the circumcircle of the triangle $XYZ$ . Then
$\angle YOZ = 2\angle YXZ = 2\angle ABC = 180^\circ - \angle BAC = 180^\circ - \angle ZAY,$
whence the quadrilateral $ZAYO$ is cyclic. The chords $OY$ and $OZ$ are equal so $\angle ZAO = \angle OAY$ . Thus $O$ lies on the bisector of the angle $BAC$ , which is the perpendicular bisector of the segment $XW$ , therefore $OW = OX = OY = OZ$ and $W$ lies on the circumcircle of the triangle $XYZ$ .

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