Tangent Trapezoid (Posted on 2010-07-28)

Let (P,r) denote a circle with center P and radius r.

Circles (A,a) and (B,b) are externally tangent with 
a less than b.

Let CD and EF be the common external tangents to
circles (A,a) and (B,b) with C and F on circle (A,a) 
and D and E on circle (B,b).

What is the area of trapezoid CDEF in terms of 
a and b?

	Submitted by Bractals
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Solution:	(Hide)
Let [KLM...XYZ] denote the area of polygon KLM...XYZ. Let the common internal tangent intersect CD and EF at points P and R respectively. Let Q be the intersection of AB and PR. Tangents to a circle from an external point have the same length. Therefore, |PC| = |PQ| = |PD| and |RE| = |RQ| = |RF| By symmetry, |PQ| = |RQ|. Therefore, |CD| = |PR| = |EF| Construct PS perpendicular to EF with S on EF and AG perpendicular to BE with G on BE. Triangles PSR and AGB are similar since corresponding sides are perpendicular to each other. Therefore, |PS| |AG| ------ = ------. |PR| |AB| Since |AG| = |EF|, we have |EF|2 = |AB|2 - |BG|2 = (b + a)2 - (b - a)2 = 4ab and |PS| = |EF|2/|AB| = 4ab/(a + b) See first |CP| |ER| ------ = ------, implies |PD| |RF| [CDEF] = [PEF] + [RCD] = 2[PEF] = |PS||EF| = 8ab*sqrt(ab)/(a + b). ----------------------------------------------------------- See Daniel's post for an alternate solution.

	Subject	Author	Date
	analytical solution	Daniel	2010-07-28 16:26:33