Question

in the cyclic quadrilateral show that the sum of the products of opposite sides is equal to product of the diagonals​

Answer 1

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Explanation:

Let us consider a cyclic quadrilateral ABCD.

To prove: (AB × CD) + (BC × AD) = AC × BD

Construction: Draw ∠ADE = ∠BDC

Proof :

∠DAC = ∠DBC are made by the same arc DC.

⇒ ∠DAC = ∠DBC  (Angles in a same segment are equal)

Consider ∆AED and ∆BCD,

∠ADE = ∠BDC (by construction)

∠DAC = ∠DBC (proved above)

⇒ ∆AED ~ ∆BCD (by AA similarity)



Similarly, we can prove that ∆ABD ~ ∆ECD



adding (1) and (2)

(AB × CD) + (AD × BC) = BD × CE + BD × AE = BD (CE + AE) = BD × AC

i.e., (AB × CD) + (AD × BC) = AC × BD