perplexus.info :: Geometry : Four Congruent Circles

Four Congruent Circles (Posted on 2006-06-14)

Show how to construct four congruent circles, inside an acute triangle ABC, with centers A', B', C', and D' such that

circle with center A' is tangent to sides AB and AC,
circle with center B' is tangent to sides BC and BA,
circle with center C' is tangent to sides CA and CB, and
circle with center D' is externally tangent to the other three circles.

Submitted by Bractals

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Solution: (Hide)

If the radius of the four circles can be constructed, then clearly the four circles can be constructed. A rough sketch of the solution shows that triangle A'B'C' is similar to triangle ABC, its incenter is the incenter of triangle ABC, and D' is its circumcenter. Let x be the radius of the four circles, R' and r' the circumradius and inradius respectively of the triangle A'B'C', and R and r the circumradius and inradius respectively of the triangle ABC. Therefore,

r'/R' = r/R,

r' = r - x, and

R' = 2*x

Solving these equations for x gives

x = R*r/(R+2*r)

The circumradius and inradius of triangle ABC can be constructed. The radius x is constructable from R and r.

Eric's "Think BIG then small... like Alice" has a concise way of constructing this radius.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
re: Think BIG then small... like Alice	brianjn	2006-06-15 08:14:16
Think BIG then small... like Alice	Eric	2006-06-14 23:43:16
Short cut - but not a construction	brianjn	2006-06-14 19:58:40
re: If this is what is meant then this is a solution	Bractals	2006-06-14 18:33:02
If this is what is meant then this is a solution	Charlie	2006-06-14 15:50:31

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