Notes
circle inside a triangle solution

Solution to the Circle Inside a Triangle Puzzle

Solution by Area of a Triangle

Decomposing the triangle as above shows that its area can be written as:

The area can be established either using Heron's formula or Pythagoras' theorem.

Using Heron's formula, the area is written in terms of the semi-perimeter which is $\frac{15 + 14 + 13}{2} = 21$.

To use Pythagoras' theorem, mark a point $D$ directly below $C$ on the edge $A B$ so that $A D C$ forms a right-angled triangle.

Applying Pythagoras' theorem to the two right-angled triangles $A D C$ and $C D B$ yields:

So $x = \frac{15^2 + 13^2 - 14^2}{30} = \frac{33}{5}$ and $y = \frac{56}{5}$. The area of the triangle is therefore