Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.3 AIME, harder Prove it United States

Problem:

Let $ABC$ be a triangle with circumcircle $\Gamma$ , whose incircle touches $BC$ , $CA$ , $AB$ at $D$ , $E$ , $F$ . We draw a circle tangent to segment $BC$ at $D$ and to minor $\operatorname{arc} \widehat{BC}$ of $\Gamma$ at the point $A_{1}$ . Define $B_{1}$ and $C_{1}$ in a similar way. Prove that lines $A_{1}D$ , $B_{1}E$ , $C_{1}F$ are concurrent.

Solution

Solution:

By the so-called "shooting lemma" (which is proved by taking homothety at $A_{1}$ ) we find that line $A_{1}D$ passes through the arc midpoint of $\widehat{BAC}$ of $\Gamma$ ; denote this arc midpoint by $X$ . Define $Y$ and $Z$ similarly, so that $Y$ lies on line $B_{1}E$ and $Z$ lies on line $C_{1}F$ .

We note the triangles $XYZ$ and $DEF$ are homothetic, since their corresponding sides are parallel: line $YZ$ and $EF$ are both known to be perpendicular to the internal $\angle A$ -bisector. Thus $DX$ , $EY$ , $FZ$ meet at a point—which thus is also the concurrency point of $A_{1}D$ , $B_{1}E$ , $C_{1}F$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.