Notes
three right-angled triangles solution

Three Right-Angled Triangles

Solution by Area of a Triangle and Corresponding Angles

In the above diagram, the point labelled $H$ is such that $H F$ is parallel to $A D$.

Consider the two triangles $G F A$ and $E D A$. The lengths of $G F$ and $E D$ are given to be the same, and the “height” of $A$ above these two “bases” is the same, so their area is the same. As triangles $G H F$ and $E C D$ are both right-angled, have the same length hypotenuse, and angles $G \hat{F} H$ and $E \hat{D} C$ are corresponding angles, these two triangles are congruent and so have the same area. Therefore triangles $E A C$ and $H F A$ have the same area.

Quadrilateral $F H A B$ is a rectangle since $H F$ is parallel to $A B$ and angle $H \hat{A} B$ is a right-angle, so triangles $F H A$ and $F B A$ have the same area.

Therefore triangles $F B A$ and $E C A$ have the same area. The triangle $I B A$ is their common overlap, so the regions $F I A$ and $E I B C$ have the same area. Therefore the pink region has area $7$.