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A quadrilateral ABCD is drawn to circumscribe a circle (see Fig.). Prove that AB + CD = AD + BC

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A quadrilateral ABCD is drawn to circumscribe a circle (see Fig.). Prove that AB + CD = AD + BC

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From this figure we can conclude a few points which are:

(i) DR = DS

(ii) BP = BQ

(iii) AP = AS

(iv) CR = CQ

Since they are tangents on the circle from points D, B, A, and C respectively.

Now, adding the LHS and RHS of the above equations we get,

DR + BP + AP + CR = DS + BQ + AS + CQ

By rearranging them we get,

(DR + CR) + (BP + AP) = (CQ + BQ) + (DS + AS)

By simplifying,

AD + BC = CD + AB

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