Notes
nested isosceles triangles solution

Nested Isosceles Triangles

Solution by Triangle area, Angle in a Semi-Circle, and Angles on a Straight Line

With the points labelled as in the above diagram, triangles $A E D$ and $D E B$ have the same area and the same height above the line $A B$, so the lengths of their bases along that line must be the same. That is, $A D$ and $D B$ have the same length. So a circle centred on $D$ that passes through $B$ will also pass through $E$ and $A$. The line segment $A B$ is then a diameter of that circle and $E$ a point on its circumference, so since the angle in a semi-circle is $90^\circ$, angle $A \hat{E} B$ is $90^\circ$. Then as angles at a point on a straight line add up to $180^\circ$, angle $B \hat{E} C$ is also $90^\circ$.