Problem 2. Let ABC be a triangle with AC > AB , and denote its circumcircle by Ω and incentre by I . Let its incircle meet sides BC, CA, AB at D, E, F respectively. Let X and Y be two points on minor arcs and of the incircle, respectively, such that ∠ BXD = ∠ DYC . Let line XY meet line BC at K . Let T be the point on Ω such that KT is tangent to Ω and T is on the same side of line BC as A . Prove that lines TD and AI meet on Ω.
Solution 1
Invert about a circle centred at with radius , followed by a flip about . The incircle is sent to a parallel line to , and is fixed by Power of a Point. Now the angle condition is reformulated as , but since lie on a parallel line to , is an isosceles trapezium.
Now as the center of lies on the perpendicular bisector of as it is a chord, by symmetry is the reflection of about the midpoint of . In fact, being tangent to , implies is the arc midpoint of as the tangency point is fixed about a reflection across the perpendicular bisector of (Else we would have two intersection points of with ), and swap under this reflection.
In fact as is fixed , but as it is the arc midpoint by the Incenter-Excenter Lemma are collinear and we are done,
Solution 2
Let , , and .
We first note that quadrilateral is cyclic, since . So, also let .
Since is tangent to the incircle of and is tangent to at , from the power of point theorem we get
and so .
From triangles and we have
By the ratio lemma, we have
and
Hence , and so,
Also, triangles and are similar, since , so
Therefore, bisects angle , and so and , as the angle bisector of , meet at the midpoint of arc .