Tangential Quadrilateral - Tangency Chords

Tangency Chords

If e, f, g and h are the tangent lengths of a tangential quadrilateral, then the lengths of the tangency chords are

where the tangency chord of length k connects the sides of lengths a = e + f and c = g + h, and the one of length l connects the sides of lengths b = f + g and d = h + e. The squared ratio of the tangency chords satisfy

The two tangency chords

• are perpendicular if and only if the tangential quadrilateral also has a circumcircle (it is bicentric).
• have equal lengths if and only if the tangential quadrilateral is a kite.

The tangency chord between the sides AB and CD in a tangential quadrilateral ABCD is longer than the one between the sides BC and DA if and only if the bimedian between the sides AB and CD is shorter than the one between the sides BC and DA.

If tangential quadrilateral ABCD has tangency points W on AB and Y on CD, and if tangency chord WY intersects diagonal BD at M, then the ratio of tangent lengths equals the ratio of the segments of diagonal BD.