May 2013
Problem of the Month

Three Circles

Three equal circles are drawn (see diagram).
A straight line connects the centers of the three circles.
Angle B is a right angle

What is the length of segment DE in terms of segment BC?

Hint: Let BC = 1, then solve for DE.

Solution to the Problem:

DE = 8/5 BC.




Dennis Beck sent in a different way to solve it using trigonometry:

angle DAF=arcsin(1/5)

angle ADF=180-arcsin(3/5)

angle FDE=arcsin(3/5)

angle DFE=arcsin(180-2 angle FDE)

DF=EF=1

Using the law of sines

Sin(angle FDE)/1=sin(angle DFE)/DE

DE=sin(angle DFE)/sin(angle FDE)

Sin(angle FDE)=3/5

Cos(angle FDE)=sqrt(1-(3/5)^2)=sqrt(1-9/25)=sqrt(16/25)=4/5

Sin(angle DFE)=sin(180-2 angle FDE)=sin(2 angle FDE)=2sin(angle FDE)cos(angle FDE)=2(3/5)(4/5)=24/25

DE=(24/25)/(3/5)=8/5=1.6

So DE=8/5(BC)

Correctly solved by:

1. James Alarie	Flint, Michigan
2. Brian Morrison	Thornton Academy, Saco, Maine
3. Dennis Beck	Clayton Valley Charter High School, Concord, California
4. Vikash Kumar	New Delhi, India

Send any comments or questions to: David Pleacher