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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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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!

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)

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