Circle Square Question

Geometry Level 5

Given 2 circles and a square, what is the greatest amount of intersections possible?


The answer is 18.

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Rmflute Shrivastav by Rmflute Shrivastav , 19, USA

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1 solution

A circle can intersect a square at 8 pints, two at each side. Two with slightly different radii, can therefore intersect at 16 points. If the circles are not concentric, they too intersect at 2 more points. So total 18 .

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any theoritical proof of the above question

akarsh jain - 4 years, 3 months ago

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Simple observation will show this is the case.
We can show by taking S as the side of the square, R and r the radii, and
S<2R, S=2R, S>2R and/or S<2r, S=2r, S>2r. Locating (0,0) at the center of the square. And for various location of Centers. But than too we have to depend on observation, and see when there is no intersection, 2, 4 and 8
per circle.
As per question given, one of the infinite possibility is as follows,


Both, the center of the square and center of circle with radius R both at (0,0);
The center of the other circle at a distance of .1R from (0,0), and r=1.1R.
But S<2r, 2R< 2 \sqrt2 S.
Theoretical proof might be very long . However this might be satisfactory.

Niranjan Khanderia - 4 years, 2 months ago

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