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Question
If the sides of a quadrilateral touch a circle, prove that the sum of a pair of opposite sides is equal to the sum of the other pair.
If the sides of a quadrilateral touch a circle, prove that the sum of a pair of opposite sides is equal to the sum of the other pair.
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Solution
Look at the diagram given below,
Each side is a sum of tangents to the incircle from two vertices of the quadrilateral. We know that tangents from one point to any circle are always equal.
AB + CD = w + x + y + z
BC + DA = x + y + z + w
Hence sum of opposite sides is equal to the sum of the other pair
Look at the diagram given below,
Each side is a sum of tangents to the incircle from two vertices of the quadrilateral. We know that tangents from one point to any circle are always equal.
AB + CD = w + x + y + z
BC + DA = x + y + z + w
Hence sum of opposite sides is equal to the sum of the other pair
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