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# Previous Puzzles of the Month + Solutions

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Puzzle # 130

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

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Source of the puzzle:
© G. Sarcone.
You cannot reproduce any part of this page without prior written permission.
 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, then:      |DG| = s – 2r = 2s - 2R so:      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 or        4(3R - 8r)(R - r) = 0 Taking the solution where R > r:        3R = 8r or 3R/8 = r Q.E.D. The 5 Winners of the Puzzle of the Month are: Arthur Vause, U.K. - Marc Hallemans, Belgium - Tony Garcia, Dominican Republic - David Almeida, Portugal - Yongting Chen, Canada Congratulations!
 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) Discuss the problem on our FaceBook page! © 2012 G. Sarcone, www.archimedes-lab.org 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. 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!
 Previous puzzles of the month...
 Puzzle 129: Steam locomotive (2012) Puzzle 128: Xmas tree (2012) Puzzle 127: Square vs Annulus (2011) Puzzle 126: A troublesome sequence (2011) Puzzle 125: A mathematical shield (2011) Apr-May 2010: A chopping problem Oct-Nov 09: The Mark of Zorro July-Sept 09: radiolarian's shell May-June 09: circle vs square Jan-Feb 09: geometric mouse Sept-Oct 08: perpendicular or not... Puzzle Archive
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