Question

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 $\angle ACP=\angle QCD$.

Answer 1

$\Rightarrow$Chords $AP$ and $DQ$ are joined.

$\Rightarrow$For chord $AP$,

$\Rightarrow$$\angle PBA=\angle ACP$ ...Angles in the same segment --- (i)

$\Rightarrow$For chord $DQ$,

$\Rightarrow$$\angle DBQ=\angle QCD$ ...Angles in same segment --- (ii)

$\Rightarrow$$ABD$ and $PBQ$ are line segments intersecting at $B$.

$\Rightarrow$$\angle PBA=\angle DBQ$ ...Vertically opposite angles --- (iii)

By the equations (i), (ii) and (iii),

$\angle ACP=\angle QCD$