Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.1 AIME, harder Prove it Saudi Arabia

Let $(O_{1})$ , $(O_{2})$ be given two circles intersecting at $A$ and $B$ . The tangent lines of $(O_{1})$ at $A$ , $B$ intersect at $O$ . Let $I$ be a point on the circle $(O_{1})$ but outside the circle $(O_{2})$ . The lines $IA$ , $IB$ intersect circle $(O_{2})$ at $C$ , $D$ . Denote by $M$ the midpoint of $CD$ . Prove that $I$ , $M$ , $O$ are collinear.

Solution

Denote $N$ as the midpoint of $AB$ . Because $A$ , $B$ , $C$ , $D$ belong to the same circle, then we have
$\triangle IAB \sim \triangle IDC.$
Since $M$ is the midpoint of $CD$ and $N$ is the midpoint of $AB$ then $IM$ , $IN$ are isogonal conjugate with respect to the angle $\angle CID$ .

In the other hand, $O$ is the intersection of two tangent lines of $(O_{2})$ at $A$ , $B$ , then $IO$ is the symmedian of triangle $IAB$ . It means $IO$ , $IN$ are isogonal conjugate with respect to $\angle AIB$ .

Hence, $I$ , $M$ , $O$ are collinear.

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