The Green Ratio

Geometry Level 3

Two quarter-circles and a semi-circle are inscribed in a unit square, as shown in the figure below. The smaller green circle is tangent to the two quarter-circles and to the top of the square. The larger green circle is tangent to the two quarter-circles and to the semi-circle.

The diameter ratio of the two green circles can be expressed as A B \dfrac A{B} , where A A and B B are coprime positive integers. Find A + B A+B .


The answer is 11.

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W Rose by W Rose , 63, USA

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1 solution

Michael Mendrin
Nov 14, 2016

Let r 1 , r 2 {r}_{1}, {r}_{2} be the radii of the small and large green circles. Then from the graphic, we have the following equations to solve separately

( 1 2 ) 2 + ( 1 − r 1 ) 2 = ( 1 + r 1 ) 2 { \left( \dfrac { 1 }{ 2 } \right) }^{ 2 }+{ \left( 1-{ r }_{ 1 } \right) }^{ 2 }={ \left( 1+{ r }_{ 1 } \right) }^{ 2 }
( 1 2 ) 2 + ( 1 2 + r 2 ) 2 = ( 1 − r 2 ) 2 { \left( \dfrac { 1 }{ 2 } \right) }^{ 2 }+{ \left( \dfrac { 1 }{ 2 } +{ r }_{ 2 } \right) }^{ 2 }={ \left( 1-{ r }_{ 2 } \right) }^{ 2 }

with solutions r 1 = 1 16 {r}_{1}=\dfrac{1}{16} and r 2 = 1 6 {r}_{2}=\dfrac{1}{6} , so that the ratio is 3 8 \dfrac{3}{8} and the answer is 11 11

10 Helpful 0 Interesting 0 Brilliant 0 Confused

Oh wow. Good observation to find those 2 equations, and that's all that we need!

Calvin Lin Staff - 4 years, 7 months ago

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Whenever you see an extraordinarily "neat" answer to a problem that seems to require a complicated solution, a simpler solution probably exists.

Michael Mendrin - 4 years, 6 months ago

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