[pohoatza]

Let be a triangle with . Let , be points on the sides and , such that the angles and are congruent. If lies in the interior of the quadrilateral such that the circumcircle of triangle is tangent to the circumcircle of and the circumcircle of is tangent to the circumcircle of , prove that the points , , , are concyclic.

Author: Cosmin Pohoata

[The QuattoMaster 6000]

pohoatza wrote:

Let be a triangle with . Let , be points on the sides and , such that the angles and are congruent. If lies in the interior of the quadrilateral such that the circumcircle of triangle is tangent to the circumcircle of and the circumcircle of is tangent to the circumcircle of , prove that the points , , , are concyclic.

Solution

Invert around , which maps point to . Now, since the circumcircles of and are tangent, we see that . Similarly, , so is a parallelogram. This means that . Also, the inversion maps . Similarly,soAlso,Which implies that , so , , and are collinear. Now, notice that is the intersection between and , so is the intersection between the circumcircles of and . Now,so , which means that , , and are collinear, so is cyclic.

[plane geometry]

Could anyone solve it without using "inversion"?

[Leonhard Euler]

From fact that circumcircle of triangle is tangent to the circumcircle of triangle , we get and similarly . By adding,
. Hence are collinear. So . Comparing with , we get which imply are concyclic.