Three Incircles

We note that $\triangle ABC$ , $\triangle CED$ and $\triangle DEA$ are similar. This means that comparing among three triangles, the area of a triangle is directly proportional to the square of any of its side lengths, since the three side lengths are in a fixed proportion. Similarly, the area of the incircle in the triangle is also directly proportional to the square of any of the side length.

Let us take the hypotenuse of each of the right triangle. Then the area of incircle in $\triangle ABC$ , $[\bigcirc_{\color{#20A900}ABC}] \propto \overline{AC}^2$ $\implies [\bigcirc_{\color{#20A900}ABC}] = k \overline{AC}^2$ , where $k$ is a constant. Similarly, $[\bigcirc_ {\color{#D61F06}CED}] = k \overline{DC}^2$ and $[\bigcirc_ {\color{#D61F06}DEA}] = k \overline{AD}^2$ .

By Pythagorean theorem , we have:

$\begin{aligned} \overline{DC}^2 + \overline{AD}^2 & = \overline{AC}^2 \\ k\overline{DC}^2 + k\overline{AD}^2 & = k\overline{AC}^2 \\ [\bigcirc_ {\color{#D61F06}CED}] + [\bigcirc_ {\color{#D61F06}DEA}] & = [\bigcirc_{\color{#20A900}ABC}] \\ [{\color{#D61F06}\text{Red}}] & = [{\color{#20A900}\text{Green}}] \end{aligned}$

Answer: They are the same.

The three triangles ABC, CED and DEA are similar. So in each of them, the inscribed circle covers the same percentage* of the triangle: $\bigcirc_{\color{#20A900}ABC} = k\% \triangle_{\color{#20A900}ABC}$ , $\bigcirc_{\color{#D61F06}CED} = k\% \triangle_{\color{#D61F06}CED}$ , $\bigcirc_{\color{#D61F06}DEA} = k\% \triangle_{\color{#D61F06}DEA}$ . But (by simmetry) we know that the areas of the triangles add up like this: $\triangle_{\color{#20A900}ABC} = \triangle_{\color{#D61F06}CED} + \triangle_{\color{#D61F06}DEA}$ . Multiplying this by k%, we get: $\bigcirc_{\color{#20A900}ABC} = \bigcirc_{\color{#D61F06}CED} + \bigcirc_{\color{#D61F06}DEA}$ .

* There is a neat proof for this, based on dimensional analysis: the ratio of the said areas is dimensionless; a dimensionless quantity cannot depend on a length -- otherwise it would have length dimensions (maybe raised to some power); so the said ratio does not vary with the triangle's size; so that ratio is the same for all similar triangles. (Actually, this is a proof that all dimensionless ratios are the same for similar figures/bodies: ratios of two lengths, ratios of two areas, ratios of two volumes, as well as ratios like volume / (area * length), area / (length * length) etc.)