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In the above figure,the circles with cen...


In the above figure,the circles with centres P,Q and R intersect at point B,C,D and E as shown. Lines CB and ED intersect in point M. Lines drawn from point M touch the circles at points A and F. Prove that MA = MF.

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In circle with centre P, MA is a tangent and MC is a secant.
`:. MA^(2) = MB xx MC` …(Tangent secant segments theorem)
In circle with centre Q,MC and ME are secant
`:. MB xx MC = MD xx ME`… (Theorem of external division of chords) ….(II)
In circle with centre R, MF is a tangent and ME is a secant.
`:. MF^(2) = MD xx ME` ... (Tangent secant segments theorem)
`:. MA^(2) = MF^(2)`.... (From I, II and III)
`:. MA = MF` ....(Taking square root)
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