3 Tutor System
Starting just at 265/hour

# Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see figure). Prove that $$\angle{ACP}$$ = $$\angle{QCD}$$.

Given:
Two circles intersect at two points B and C. Through B two line segment ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively.

To prove: $$\angle{ACP}$$ = $$\angle{QCD}$$

Proof:
In circle I,
$$\angle{ACP}$$ = $$\angle{ABP}$$ ...(i)
(Angles in the same segment)

In circle II,
$$\angle{QCD}$$ = $$\angle{QBD}$$ ...(ii)
(Angles in the same segment)

Also, $$\angle{ABP}$$ = $$\angle{QBD}$$
(Vertically opposite angles)

From Equation (i) and (ii), we get,

$$\angle{ACP}$$ = $$\angle{QCD}$$
Hence, proved.