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China Western Mathematical Olympiad

Geometry Difficulty 5.4 AIME, harder Prove it China

$AB$ is a diameter of the circle $O$ , the point $C$ lies on the extended line $AB$ produced. A line passing through $C$ intersects with the circle $O$ at points $D$ and $E$ . $OF$ is a diameter of the circumcircle $O_1$ of $\triangle BOD$ . Join $CF$ and its extension, which intersects the circle $O_1$ at $G$ . Prove that points $O$ , $A$ , $E$ , $G$ are concyclic.

Solution

Thus, $G$ , $A$ , $C$ , $D$ are concyclic. Hence,
$\angle AGC = \angle ADC, \qquad ①$
$\angle AGC = \angle AGO + \angle OGF = \angle AGO + \frac{\pi}{2}, \qquad ②$
$\angle ADC = \angle ADB + \angle BDC = \angle BDC + \frac{\pi}{2}. \qquad ③$
Combining ①, ② and ③ yields
$\angle AGO = \angle BDC. \qquad ④$
Since $B$ , $D$ , $E$ , $A$ are concyclic, we have
$\angle BDC = \angle EAO. \qquad ⑤$
As $OA = OE$ , we obtain
$\angle EAO = \angle AEO. \qquad ⑥$
Combining ④, ⑤ and ⑥ implies $\angle AGO = \angle AEO$ .
Therefore, $O$ , $A$ , $E$ , $G$ are concyclic.

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