AoPS - Problem

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Problem

Source: Greece JBMO TST 2017, Problem 2

Tags: geometry, Greece

Amir Hossein

26.06.2018 00:35

Let $ABC$ be an acute-angled triangle inscribed in a circle $\mathcal C (O, R)$ and $F$ a point on the side $AB$ such that $AF < AB/2$ . The circle $c_1(F, FA)$ intersects the line $OA$ at the point $A'$ and the circle $\mathcal C$ at $K$ . Prove that the quadrilateral $BKFA'$ is cyclic and its circumcircle contains point $O$ .

Yaghi

26.06.2018 14:54

Note that $A'FB=180-2C$ which means that $FBOA'$ is cyclic.also $FK=FA,OK=OA \implies FKO=FAO=90-C=FBO$ so $BKFO$ is cyclic as well and we are done.

zuss77

28.06.2018 11:00

This problem badly needes to be reformulated: Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ are intersecting at points $A$ and $B$ . Line $AO_1$ intersects $\omega_2$ at $C$ . Line $AO_2$ intersects $\omega_1$ at $D$ . Prove that points $B, C, D, O_1, O_2$ lay on circle.

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