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’.

Question

Philoid · Accepted Answer

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.