Let M and N be the midpoints of segments BC and AC of a triangle ABC, respectively. Let Q be a point on the line through N parallel to BC such that Q and C are on opposite sides of AB and ∣QN∣⋅∣BC∣=∣AB∣⋅∣AC∣.Suppose that the circumcircle of triangle AQN intersects the segment MN a second time in a point T=N.
Prove that there is a circle through points T and N touching both the side BC and the incircle of triangle ABC. geometrycircumcircleincircletouching circlessidelengthsgeometry proposed