Question

Let be a circle with centre $O$. A second circle $L$ passes through $O$ and cuts $K$ at two other points $A$ and $B$. Let $P$ be a point on arc $AOB$ and extend line segment $BP$ to intersect $K$ at a second point $Q$. Prove that the bisector of angle $APQ$ passes through $O$.

Answer 1

The bisector of angle APQ passes through O and this is illustrated below.

How to illustrate the information?

From the information given, the center is O. and the circle passes through O and cuts at K.

In this case, it should be noted that the circles are equal according to the SAS test.

Here, AOB + APQ = 180° (Linear pair)

2AOB = 180

AOB = 90.

Therefore, the bisector of angle APQ passes through O.

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