Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.5 AIME, harder Prove it

Example 1 As shown in Figure 1, in $\square A B C D$ , $\odot O$ passes through points $A, B, C$ , and $E$ is a point on $B C$ . Let the circumcircle of $\triangle A B E$ be $\odot P$ , $P F \perp B C$ , and $P F$ intersects $A B$ at point $F$ . $C F$ intersects $\odot O$ at point $G$ . Prove: $G, E, C, D$ are concyclic.

Solution

【Analysis and Proof】As shown in Figure 1, extend $EF$ to intersect $\odot P$ at point $H$ , and connect $HG$ , $HA$ .
Since $PF \perp BC$ , by symmetry we know
$AH \parallel BE$
$\Rightarrow H, A, D$ are collinear $\Rightarrow EH=BA=CD$
$\Rightarrow$ Quadrilateral $HECD$ is an isosceles trapezoid
$\Rightarrow H, E, C, D$ are concyclic.
By the intersecting chords theorem, we have
$FH \cdot FE = FA \cdot FB = FG \cdot FC \text{.}$

Therefore, $H, G, E, C$ are concyclic.
Thus, $H, G, E, C, D$ are concyclic.
Hence, $G, E, C, D$ are concyclic.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.