Three Tangent Circles

Three mutually tangent circles of radius one are surrounded by a larger circle that is simultaneously tangent to all three. What is the radius of the larger circle?
Source: www.mathcircles.org

SOLUTION

The centers A,B,C of the three mutually tangent circles form an equilateral triangle ABC with side = 2. The altitudes are also medians and angle bisectors (not all are drawn for clarity). They all intersect at point O.
Length of altitude:
AH=\sqrt{3}
Ratio of median:
OA:AH=2:3

\frac{OA}{AH}=\frac{2}{3}
\frac{OA}{\sqrt{3}}=\frac{2}{3}
OA=\frac{2\sqrt{3}}{3}

Radius of larger circle
OP=OA+AP
=\frac{2\sqrt{3}}{3}+1
=\frac{2\sqrt{3}+3}{3}

Answer: \frac{2\sqrt{3}+3}{3}

Advertisements

About mvtrinh

Retired high school math teacher.
This entry was posted in Problem solving and tagged , , , , , , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s