Right-angled triangle divided to two triangles with equal inradii

Let ABC is the triangle with right angle at vertex C.
Construct the point D on hypotenuse AB such that
two triangles, BDC and DAC have the equal inradii.

Let the side CB is less than side CA.

1. Take two points on side CA: F and E, such that:
   CF = min{CA/2, CB}, and CE = max{CA/2, CB}
2. Find the point O, the midpoint of CE
3. Draw the circle with center at point O and radius CO = OE
4. From point F draw perpendicular to side CA
5. Find point H of intersection of line in p. 4 and circle in p. 3
6. Draw the circle with center at vertex C and radius CH
7. Find the point D of intersection of circle in p. 6 with the hypotenuse BA.

The segment CD divides the triangle ABC into two triangles, BDC and DAC
with incircles of equal radii.