Cyclic Quadrilateral - Diagonals

Diagonals

In a cyclic quadrilateral with successive vertices A, B, C, D and sides a = AB, b = BC, c = CD, and d = DA, the lengths of the diagonals p = AC and q = BD can be expressed in terms of the sides as

and

According to Ptolemy's second theorem,

using the same notations as above.

For the sum of the diagonals we have the inequality

Equality holds if and only if the diagonals have equal length, which can be proved using the AM-GM inequality.

In any convex quadrilateral, the two diagonals together partition the quadrilateral into four triangles; in a cyclic quadrilateral, opposite pairs of these four triangles are similar to each other.

If M and N are the midpoints of the diagonals AC and BD, then

where E and F are the intersection points of the extensions of opposite sides.