Let ABC be a 3-4-5 triangle with right angle C. Let D be a point on the hypotenuse. CD then partitions ABC into ACD and BCD.

Where is D located if ACD and BCD have congruent incircles?

Construction of point D for any right triangle. 

If the triangle is isosceles, then D is the 
midpoint of the hypotenuse; otherwise, label
the vertex opposite the shorter leg A and the
other vertex B.

Construct point E on line BC such that
|CE| = |AC|/2 and vertex C lies between B and E.

Label the midpoint of line segment BE as M.

Construct point F on side AC such that |MF| = |ME|.

The circle with center C and radius |CF|
intersects the hypotenuse AB at two points.
The point closest to vertex A is point D.  

Posted by Bractals on 2017-01-07 12:15:30