Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 6.3 National Olympiad Prove it Estonia

In a plane there is a triangle $ABC$ . Line $AC$ is tangent to circle $c_A$ at point $C$ and circle $c_A$ passes through point $B$ . Line $BC$ is tangent to circle $c_B$ at point $C$ and circle $c_B$ passes through point $A$ . The second intersection point $S$ of circles $c_A$ and $c_B$ coincides with the incenter of triangle $ABC$ . Prove that the triangle $ABC$ is equilateral.

Figure 5

Solution

By the tangent-secant theorem we have $\angle BCS = \angle CAS$ and $\angle ACS = \angle CBS$ (see fig. 5). The incenter of a triangle is the point of intersection of angle bisectors, therefore $\angle CAB = 2\angle CAS = 2\angle BCS = \angle BCA$ and $\angle CBA = 2\angle CBS = \angle ACS = \angle BCA$ . Hence $ABC$ is equilateral.

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