Consider a right triangle with an inscribed circle. Let x and y be the lengths of the two line segments formed on the hypotenuse by the point of tangency with the circle. What interesting fact can you prove about x*y?
Let A be the right angle, and O the center of the circle, of radius
R. Let the circle be tangent to AB at L; to AC at M; and to BC at
N. Let BN=X and CN=Y; X+Y=BC=Z.
Since BO bisects angle B, BL=BN, so
AB=X+R. For a similar reason, AC=Y+R. Since AB²+AC²=BC², we have
(X+R)²+(Y+R)²=(X+Y)². Simplifying, RX+RY+R²=XY, so XY=R²+RZ.
A possible solution
