Proposed Problem

Click the figure below to see the complete problem 336 about Two equal circles, a Common Tangent and a Square.

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Complete Problem 336

Level: High School, SAT Prep, College geometry

## Friday, August 7, 2009

### Problem 336. Two equal circles, a Common Tangent and a Square

Labels:
circle,
common tangent,
square

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Extend FM to meet AC in N. Also extend DA & EB to meet the two circles in D' & E' respectively.

ReplyDeleteNow, AN = r - x/2 & NM = r - x. From triangle MAN, (r-x/2)^2+(r-x)^2=r^2 from which we get x=2r/5 or x=2r. The former is the side of square FGHM while the latter is the side of DEE'D'

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Extend MH so that it meets the radii AD and BE at points P and Q

ReplyDeleteLet PM = HQ = y

Then we have two equations:

(r-x)^2 + y^2 = r^2 ..... (1)

and

2y + x = 2r ..........(2)

Eliminate y between the two equations to get a quadratic which resolves to

x = (2/5)*r or x = 2r.

x = 2r is absurd.

So x = (2/5)*r is the answer

which will