## Inscribed Circle of Equilateral Triangle

The circumference of a circle circumscribing an equilateral triangle is $24\pi$ units. Find the number of square units in the area of the circle inscribed in the same triangle.
Source: NCTM Mathematics Teacher, February 2006

SOLUTION

In equilateral triangle $ABC$, angle bisector $BM$ is also the perpendicular bisector of $AC$. That makes point I the circumcenter, the incenter, and the centroid of triangle $ABC$.
$24\pi=2\pi\times IB$
$IB=12$
$IM=IB/2=6$ because $I$ is the centroid
Area of incircle
$\pi\times IM^2=36\pi$

Answer: $36\pi$ square units