Ceva in Circumscribed Quadrilateral

Let ABCD be a quadrilateral circumscribed around a circle. Denote the lengths of tangents from the vertices A, B, C, and D to the circle as a, b, c, d, respectively. Finally, let P be the point of intersection of the diagonals AC and BD. Then we have

Proof

Let E be the point of tangency of the incircle on side AB and F be the point of tangency on side BC.

By Brianchon's theorem the lines AF, BP, and CE concur at, say, point Q. Apply Ceva's theorem to ABC:

In other words,

And, finally,

Remark

Darij Grinberg has gracefully noted that the theorem has been established by more elementary means elsewhere.

References

1. R. Honsberger, More Mathematical Morsels, MAA, 1991, p. 61.

Copyright © 1996-2008 Alexander Bogomolny