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I need a little help with this geometry problem, that I need to solve synthetic (I know the analytic solution).

We write in an isosceles triangle those sides are 13 cm, 13 cm, 10 cm three circles. Every circle is tangent to two sides of the triangle and the other two circles. The radii of the circles that are tangent to the base are congruent. Find the radii of the three circles.

I also managed to prove synthetically that the two congruent radii are 2 cm.

Please help me finding the third one.

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    $\begingroup$ Hint: $(5,12,13)$ is a Pythagorean triple. If you depict the configuration on graph paper, you can get a good idea about the locations of the incenters involved. $\endgroup$ – Jack D'Aurizio Mar 22 '18 at 19:49
  • $\begingroup$ You may also consider that Steiner's construction of the Malfatti circles is pretty involved, but it greatly simplifies if the original triangle is an isosceles one. $\endgroup$ – Jack D'Aurizio Mar 22 '18 at 20:00
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This is an implementation of Steiner's construction of the Malfatti circles for an isosceles triangle.

enter image description here

  1. The red circles are symmetric and they are straighforward to locate;
  2. The orange circles are constructed in such a way that they are tangent to the "oblique" sides and they go through an incenter of a red circle and the midpoint of the incenters of the red circles;
  3. The orange circles locate the center of the blue circle and the problem is solved.
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