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All the circles and semi-circle are tangent to each other and are inscribed in a square. The 3 small circles are the same size, with radius r. R is the radius of the red circle. Prove that r = 3R/8

Difficulty level: bulbbulb, general math knowledge.
Category: Geometry.
Keywords: Square, circle, semi-circle, radius.
Related puzzles:
- Steam locomotive,
- A mathematic shield.

Source of the puzzle:
© G. Sarcone.
You cannot reproduce any part of this page without prior written permission.
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Solution to puzzle 130

Point A is the center of the red circle,
B and E are center points of small circles,
D is the center point of the semicircle,
DEF is a straight line,
EC is perpendicular to AD.

Define s = |DF|, i.e. 1/2 length of the square,
     |DG| = s – 2r = 2s - 2R

     s = 2(R - r)

Define x = |CE|

Applying Pythagoras theorem to the triangle CDE:
(1)   x2 + r2 = (s - r)2
       x2 + r2 = (2R - 3r)2
Applying Pythagoras theorem to the triangle ACE:
(2)   x2 + (R + s - 3r)2 = (R + r)2
       x2 + (3R - 5r)2 = (R + r)2
Subtracting (1) from (2):
(3)   x2 + (3R - 5r)2 - r2 = (R + r)2 - (2R - 3r)2
Rearranging (3):
       12R2 - 44Rr + 32r2 = 0
       4(3R - 8r)(R - r) = 0
Taking the solution where R > r:
       3R = 8r or 3R/8 = r

cup winnerThe 5 Winners of the Puzzle of the Month are:
Arthur Vause, U.K. UK flag - Marc Hallemans, Belgium Belgium flag - Tony Garcia, Dominican Republic Dominican Republic flag - David Almeida, Portugal Portugal flag - Yongting Chen, Canada Canada flag


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Beyond the challenge

How to Mathematically Square the Circle
There isn't any method to "geometrically" square a circle WITH compasses and triangle set squares (tough several approximation methods exist).
In the picture, the area A of the green circle equals the area A of the yellow square. The distance πR represents obviously one 1/2 circle rotation. The diameter of the semi-circle is then:
πR + R or R(π + 1)

example 2

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© 2012 G. Sarcone,
You can re-use content from Archimedes’ Lab on the ONLY condition that you provide credit to the authors (© G. Sarcone and/or M.-J. Waeber) and a link back to our site. You CANNOT reproduce the content of this page for commercial purposes.

Creative Commons License
Puzzle of the Month by Gianni A. Sarcone is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
You're encouraged to expand and/or improve this article. Send your comments, feedback or suggestions to Gianni A. Sarcone. Thanks!
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Previous puzzles of the month...
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Puzzle 129: Steam locomotive (2012)
Puzzle 128:

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Puzzle 127: Square vs Annulus (2011)
Puzzle 126: A troublesome sequence (2011)
Puzzle 125: A mathematical shield (2011)
Apr-May 2010
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Oct-Nov 09:
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July-Sept 09:
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May-June 09:
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Jan-Feb 09
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Sept-Oct 08
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July-Aug 08:
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Jan 07:
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Aug-Sept 2006
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