PROBLEM (Source=ayan.nmath)
In a convex quadrilateral the diagonals are perpendicular to each other and they intersect at Let be a point on the side such that The circle passing through and tangent to line at intersects at Prove synthetically that if then are collinear.

SOLUTION 1 (My solution): Let and . Note that and are collinear.

Also, is the perpendicular bisector of .

And, By Pascal's Theorem on , we get lies on are collinear.

Thus, As is cyclic, so is cyclic are collinear.

SOLUTION 2 (Synthetic_Potato): From power of point, we see that . Thus if is the reflection of over , we see that implies .
Let be . It suffices to now show that . Let . Now from radical axis theorem on cyclic quadrilaterals , we see that . Thus from brokard's theorem on , the polar of passes through . From the defination of Polar, . Thus and so, are collinear.