in two concentric circles, prove that all chords of the outer circle which touches the inner circle are of equal lengths.

Consider two concentric circles with centres at O. Let AB and CD be two chords of the outer circle which touch the inner circle at the points M and N respectively.

To prove the given question, it is sufficient to prove

AB = CD. For this join OM, ON, OB and OD.

Let the radius of outer and inner circles be

*R*and*r*respectively.AB touches the inner circle at M.

∴ AB is a tangent to the inner circle

∴ OM⊥AB

⇒ BM = ½ of AB

⇒ AB = 2BM

Similarly ON⊥CD, and CD = 2DN

Hence proved

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