2019 IMO Problems/Problem 6
Problem
Let be the incenter of acute triangle with . The incircle of is tangent to sides , , and at , , and , respectively. The line through perpendicular to meets again at . Line meets ω again at . The circumcircles of triangles and meet again at . Prove that lines and meet on the line through perpendicular to .
Solution
Step 1
We find an auxiliary point
Let be the antipode of on where is radius
We define cyclic an inversion with respect swap and is the midpoint Let meets again at S (other than D). We define Opposite sides of any quadrilateral inscribed in the circle meet on the polar line of the intersection of the diagonals with respect to and meet on the line through perpendicular to The problem is reduced to proving that