Given a line segment $AB$. A point $C$ is chosen on it. Three circles with the diameters $AB, BC, AC$ are constructed. A line comes through the point $C$ and intersects the large circle in the points $G$ and $J$, the smaller one in the points $H$ and $K$. How can I prove that the segments $JK$ and $GH$ are equal?
I have tried to solve the problem with the help of the similar right triangles $ACK$ and $CHB$, but after some attempts I feel like it is a dead end. What else can I do?
(I feel like I saw this problem somewhere, so apologies if this problem is popular or has been asked more than once)
With a few invocations of Thales' Theorem ...
$\square AK^\prime B H^\prime$ is an inscribed rectangle with sides parallel to $\overline{GJ}$. The result follows.