Two circles touch internally at X and a straight line cuts them at A, B, C, D in order. Prove that AB, CD subtend equal angles at X. Two circles touch internally at X and a straight line cuts them at A, B, C, D in order. Prove that AB, CD subtend equal angles at X. Source: Challenge and Thrills in Pre College Mathematics. > ![enter image description here](

![enter image description here]( Let $AD$ intersect the smaller circle at $E\
e X$ and $h$ the common tangent of the two circles at $X$. It follows that $\angle ABX=\angle CEX=180^\circ - \angle CBX$.

On the other hand, $\angle XCE=\angle XAD$ because both of them are equal to $\angle \alpha$.