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# If the Sides of a Quadrilateral Abcd Touch a Circle, Prove That: Ab + Cd = Bc + Ad - ICSE Class 10 - Mathematics

ConceptTangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments

#### Question

If the sides of a quadrilateral ABCD touch a circle, prove that:
AB + CD = BC + AD #### Solution Let the circle touch the sides AB, BC, CD and DA of quadrilateral ABCD at P, Q, R and S respectively.
Since AP and AS are tangents to the circle from external point A
AP = AS .......(i)
Similarly, we can prove that:
BP = BQ .......(ii)
CR = CQ .......(iii)
DR = DS ........(iv)
AP + BP + CR + DR = AS + DS + BQ + CQ
AB + CD = AD + BC
Hence, AB + CD = AD + BC

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Solution If the Sides of a Quadrilateral Abcd Touch a Circle, Prove That: Ab + Cd = Bc + Ad Concept: Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments.
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