2020 INMO Problems/Problem 1
Let and be two circles of unequal radii, with centres and respectively, intersecting in two distinct points and . Assume that the centre of each circle is outside the other circle. The tangent to at intersects again in , different from ; the tangent to at intersects again at , different from . The bisectors of and meet and again in and , respectively. Let and be the circumcentres of triangles and , respectively. Prove that is the perpendicular bisector of the line segment .
Let be the circumcenters of triangles , and .
Note that where follows from the tangency, proving the claim.
Now remark that , where are the centers of . By looking at the spiral similarity pivoted at , taking to , we conclude that goes to , so, the quadrilaterals and are all similar, yielding .
Finally, note that . Let be the reflection of in , clearly, are reflections in the perpendicular bisector of , thus, follows. ~anantmudgal09