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
|(1)||AP/PC = a/c.|
Let E be the point of tangency of the incircle on side AB and F be the point of tangency on side BC.
|AP/PC · CF/FB · BE/EA = 1.|
In other words,
|AP/PC · c/b · b/a = 1.|
|AP/PC = a/c.|
Darij Grinberg has gracefully noted that the theorem has been established by more elementary means elsewhere.
- R. Honsberger, More Mathematical Morsels, MAA, 1991, p. 61.
Copyright © 1996-2018 Alexander Bogomolny