convex quadrilateral problem

Question

Dear Sir or Madam,

The lenght of the sides of a convex quadrilateral, in clockwise order,are a,b,c,d. How can I prove that a,b,d, c are consecutive terms of an arthmetich progression,then the quadrilateral is tangential?

I look forward to hearing from you.

yours faithfully,

Answer 1

This is very easy. In tangential quadraliteral sums of oppposite sides are equal each other.

b - a = d - b = c - d = k (common difference). b = a + k, d = a + 2k, c = a + 3k .

a + c = 2a + 3k and b + d = 2a + 3k.

So a + c = b + d.

So if a,b,d,c is arithmetic progression then a,b,c,d sides of convex tangential quadraliteral.