Two circles with centers 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’.


Firstly draw two circles with center O and O’ such that they intersect at A and B.


Draw a line PQ parallel to OO’.


In the circle with center O, we have:


OP and OB are the radii of the circle. PB is the chord with OM as its perpendicular bisector.


i.e. BM = MP ………………………(1)


In the circle with center O’, we have:


O’B and O’Q are the radii of the circle. BQ is the chord with O’N as its perpendicular bisector.


i.e. BN = NQ ………………………(2)


From (1) and (2), we have:


BM + BN = MP + NQ


(BM + BN) + (BM + BN) = (BM + BN) + (MP + NQ)


2(BM + BN) = (BM + BN) + (MP + NQ)


2(OO’) = (BM + MP) + (BN + NQ)


2(OO’) = BP + BQ


2(OO’) = PQ


Hence, proved.


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