2008 Indonesia MO Problems/Problem 1

Problem

Given triangle $ABC$ . Points $D,E,F$ outside triangle $ABC$ are chosen such that triangles $ABD$ , $BCE$ , and $CAF$ are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.

Solution

Let $X$ be the intersection of the circumcircles of $\triangle ABD$ and $\triangle ACF$ . Note that $AXBD$ and $AXCF$ are cyclic quadrilaterals. Thus, $\angle ADB + \angle AXB = 180^\circ$ and $\angle AFC + \angle AXC = 180^\circ$ .

We know that $\triangle ABD$ and $\triangle ACF$ are equilateral, so $\angle ADB = \angle AFC = 60^\circ$ . Therefore, $\angle AXB = \angle AXC = 120^\circ$ , so $\angle BXC = 120^\circ$ .

Since $\triangle BEC$ is equilateral as well, $\angle BEC = 60^\circ$ . Note that $\angle BXC + \angle BEC = 180^\circ$ , and since the circumcircle is the circle that passes through $B, E, C$ , point $X$ must also be on the same circumcircle of $\triangle BEC$ . Thus, the cicumcircles of these three triangles are concurrent.

2008 Indonesia MO (Problems)
Preceded by First Problem	1 • 2 • 3 • 4 • 5 • 6 • 7 • 8	Followed by Problem 2
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2008 Indonesia MO Problems/Problem 1

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