I was wondering if someone could give me a hint as to how to solve this problem:
Let circle C1 intersect circle C2 at P and Q. A line intersects C1 at A and B, and C2 at C and D, such that the four points lie on the line on the order A, B, C, D and that P and Q lie on the same side of the line. Compute APC+BQD.
I found this problem on an angle chasing worksheet by Ray Li. I've tried proving APC+BQD=180°, but I'm unsuccessful because I don't know how to obtain information about the angles. If you have any ideas, please let me know.
You have requested a hint to solve this angle chasing problem. We would like to give you a hint in the form of a diagram, Here, we have extended your sketch by adding a point $E$ and four red lines, i.e., $AQ$, $AE$, $EQ$, and $PQ$. Point $E$ is the point of intersection between extended $DQ$ and the circle $\Omega_1$. If you do the angle chase using these lines, you will be able to compute $\angle APC+\angle BQD$ easily. You have also mentioned that you are trying to prove $\angle APC+\angle BQD = {\large{\pi}}$. We think that you are on track.