Competition problem — Geometry — Olympiad Maths Prep

Geometry Difficulty 5.9 AIME, harder Prove it Turkey

The points $A$ and $B$ lie on a circle with diameter $CD$ and on different sides of the line $CD$ . A circle $\Gamma$ passing through the points $C$ and $D$ intersects the line segment $AC$ at a point $E$ different from its endpoints, and the line $BC$ at a point $F$ . $P$ is the point of intersection of the tangent line to $\Gamma$ at $E$ and the line $BC$ , and $Q$ is a point different from $E$ lying on the circumcircle of the triangle $CEP$ and satisfying $QP = EP$ . $S$ is the midpoint of the line segment $EQ$ and $R$ is the point of intersection of the lines $AB$ and $EF$ . Show that the lines $DR$ and $PS$ are parallel.

Solution

As $AB$ is the Simson line for the point $D$ and the triangle $FCE$ , $DR$ is perpendicular to $EF$ . We have $\angle QEP = \angle EQP = \angle ECF = \angle XEF$ , where $X$ is a point on the ray $PE$ beyond $E$ . Therefore $Q$ , $E$ , $F$ are collinear. As $PS$ is perpendicular to $EQ$ , the result follows.

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