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Taiwan IMO Selection Camp

Geometry Difficulty 5.4 AIME, harder Prove it Taiwan

Let two lines $BC$ , $EF$ be parallel to each other, and let $D$ be a point on segment $BC$ distinct from $B$ , $C$ . Lines $BF$ , $CE$ intersect at point $I$ . Denote the circumcircles of $\triangle CDE$ , $\triangle BDF$ by $K$ , $L$ respectively. Circles $K$ , $L$ are tangent to $EF$ at points $E$ , $F$ respectively. Let $A$ be the other intersection point of circles $K$ , $L$ distinct from $D$ . Let line $DF$ meet circle $K$ again at point $Q$ , and let line $DE$ meet circle $L$ again at point $R$ . Let line $EQ$ and line $FR$ intersect at point $M$ .
Prove that $I$ , $A$ , $M$ are collinear.

Solution

1. Since $BC \parallel EF$ , and circles $K$ , $L$ are tangent to $EF$ at points $E$ , $F$ respectively, we know that points $E$ , $F$ are respectively the midpoints of arc $CDE$ , arc $BFD$ . We obtain $BF = DF$ , $CE = DE$ . Therefore,
$\angle EFD = \angle FDB = \angle DBF = \angle EFI.$
Similarly, $\angle FED = \angle FEI$ . From this we know $\triangle DEF \sim \triangle IEF$ .

2. $A$ , $E$ , $I$ , $F$ are concyclic:
$\begin{align*} \angle EAF &= \angle (AE, AD) + \angle (AD, AF) = \angle (CE, CD) + \angle (BD, BF) \\ &= \angle (CI, BC) + \angle (BC, BI) = \angle (CI, BI) \\ &= \angle (EI, FI) = \angle EIF. \end{align*}$

$\begin{align*} \angle AEQ &= \angle FEQ - \angle FEA = \angle EDQ - \angle FIA \\ &= \angle EDF - \angle FIA = \angle FIE - \angle FIA = \angle AIE. \end{align*}$

3. Let $AD$ intersect $EF$ at $N$ . Then $NF^2 = NA \cdot ND = NE^2$ . So $NF = NE$ , and $AD$ is the median of $\triangle DEF$ . Since $\triangle DEF \cong \triangle IEF$ , and $\angle ADF = \angle AEQ = \angle AIE$ , $\angle ADE = \angle AEF = \angle AIF$ , $AI$ is the $A$ -symmedian of $\triangle EIF$ .
Since $EQ$ , $FR$ are the tangent lines of $K$ at $E$ , $F$ , the $A$ -symmedian of $\triangle EIF$ passes through the intersection point $M$ of $EQ$ , $FR$ , that is, $M$ lies on line $AI$ . Q.E.D.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from zh; metadata (topic, difficulty) added by this project.