Question

A circle with diameter r is internally tangent to a circle with radius r at the point T. For some other point S on the larger circle, chord ST intersects the smaller circle at point X, and the tangents to a larger circle at S and T meet at point Y. Show that X, Y and the centre of the larger circle are collinear.

Answer 1

Let C be the center of the larger circle.
CX extended is a radius of the circle, so it is perpendicular to the chord ST, at its midpoint X.
The tangents SY and TY are the same length, so STY is an isosceles triangle, with altitude XY.
So, ST ⊥ XY, meaning C,X,Y are collinear.