Notes
triangle over a circle solution

Solution to the Triangle Over a Circle Puzzle

Posted on Feb 17, 2021

Solution by Angles in the Same Segment are Equal and Equilateral Triangles

In the above diagram, point $O$ is the centre of the circle and $E$ is the point on the circumference with the property that the lengths of $A E$ and $D E$ are equal, so that triangle $A E D$ is isosceles. Point $F$ is where the line through $E O$ extends to meet $A D$.

By the result that angles in the same segment are equal, angles $A \hat{E} D$ and $A \hat{C} D$ are equal. Since angle $A \hat{C} D$ is the interior angle of an equilateral triangle, it is $60^\circ$. So triangle $A E C$ is an isosceles triangle with an angle of $60^\circ$ and so is equilateral.

The height of an equilateral triangle is $\frac{\sqrt{3}}{2}$ of its side length, and the centre of its circumcircle is at $\frac{1}{3}$ of its height, so the length of $O E$ is $\frac{2}{3}$ of the height of triangle $A E D$ which is $\frac{\sqrt{3}}{2} \times 9$. This is the radius of the circle, and so the area of that circle is: