Question

if the sides of qiadrilateral touch a circle prove that the sum of opposite sides is equal to the sum of other pair

Answer 1

see diagram.

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.