perplexus.info :: Shapes : More Than Four Tangent Circles

More Than Four Tangent Circles (Posted on 2024-01-11)

Inside a particular isosceles triangle there are four congruent unit circles (radius=1) tangent along its base. The circles on the end are also tangent to the lateral sides of the triangle.
Then a circle of radius R is tangent to the two middle unit circles and the two lateral sides of the triangle.
Finally one more unit circle is placed atop the larger circle tangent to it and the two lateral sides.

What is the radius of the large circle?
Can you generalize for N circles along the base?

For comparison, the classic problem Four Tangent Circles is a version of this one with only two unit circles at the base.

No Solution Yet Submitted by Brian Smith

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I get an approximate solution for the radius of 2.3981117

by drawing a picture in Desmos Geometry and altering the height.

https://www.desmos.com/geometry/bsunzeoqs1

There are two competing radii as function of the height:

They are the circle that fits between the three unit circles and the concentric circle that fits in the triangle. Their equations are graphed here as functions of the height.

https://www.desmos.com/calculator/j3txwbhfja

I have not been able to solve this system. If I try again, I might make the radius the parameter and try to make it tangent to the sides of the triangle.

Posted by Jer on 2024-01-12 13:11:33

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