Two circles with centres O and O’ intersect at two points A and B. A line PQ is drawn parallel to OO’ through A (or B) intersecting the circles at P and Q.
Prove that PQ =2 OO’.
Given, draw two circles having centres O and O’ intersect at points A and 8.
Also, draw line $P Q$ parallel to $O O^{\prime}$.
Construction Join $O O^{\prime}, O P, O^{\prime} Q, O M$ and $O^{\prime} N$.
To prove $P Q=200^{\prime}$
Proof in $\Delta O P B$, $B M=M P$ [OM is the perpendicular bisector of $P B$ ]
and in $\triangle O^{\prime} B Q$, $B N=N Q$ $\left[O^{\prime} N\right.$ is the perpendicular bisector of $\left.B Q\right]$
$\therefore$ $B M+B N=P M+N Q$
$\Rightarrow \quad 2(B M+B N)=B M+B N+P M+N Q$
[adding both sides $B M+B N$ ]
$\Rightarrow$ $2 O O^{\prime}=(B M+M P)+(B N+N Q)$
$\left[\because O O^{\prime}=M N=M B+B N\right]$
$=B P+B Q=P Q$
$\Rightarrow \quad 200^{\prime}=P Q$
Hence proved.
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