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If a line l intersects two concentric circles at points A, B, C, and D as shown in the figure, prove that AB = CD. 


Prove that two concentric circles intercept equal portions on any straight line that cuts them.

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Let O be the centre of the two concentric circles and OM the perpendicular from O to the line l. AD is the chord of the larger circle and BC, the chord of the smaller circle.

Since, perpendicular from the centre to a chord bisects the chord, therefore, 

AM = MD ...(i) 

and BM = MC ...(ii)

 (i) and (ii) gives, AM – BM = MD – MC 

AB = CD.

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