Notes
triangle inside a square solution

Solution to the Triangle Inside a Square Puzzle

Solution by Angle in a Semi-Circle and Similar Triangles

In the above diagram, a second square has been placed on top of the original one. As $O A$, $O C$, and $O D$ are all the same length, the circle centred at $O$ that passes through $A$ also passes through $C$ and $D$. Since triangle $O A E$ is isosceles, it also passes through $E$.

Triangles $O E A$ and $O E D$ have the same area since their “bases” $O A$ and $O D$ have the same length and they have the same apex, $E$, above the line $A D$. So triangle $O A E$ has half the area of triangle $D A E$.

Triangle $D A E$ is right-angled since the angle in a semi-circle is $90^\circ$ and is similar to triangle $D A B$. So also is triangle $A E B$. Since the length of $A D$ is twice that of $A B$, the length of $A E$ is twice that of $E B$, and that of $D E$ is twice that of $A E$. So $D E$ has length four times that of $E B$, meaning that triangle $D E A$ is four fifths of triangle $D A B$. Hence triangle $O A E$ is two fifths of triangle $D A B$. Since triangle $D A B$ has the same area as the original square, the yellow triangle has area $\frac{2}{5}$ths of that square.