math - Check if three circles fit inside a triangle -


i have thought time writing program tells me if 3 circles given diameters can fit inside triangle given side lengths without overlapping (touching ok) each other.

how 1 think doing that?

i try find way enumerate possible configurations 3 circles, , test each configuration until 1 found 3 circles fit, or until configurations have been tested , rejected.

(in follows assuming each circle known fit in triangle itself. if circle fails fit fails fit in configuration of 3 circles.)

configuration (1) involves putting circle in each corner of triangle. (this configuration spotted.)

a triangle big green circle in top corner, medium red circle in bottom right corner, , small blue circle in bottom left corner

there 6 ways arrange circles, , each arrangement sufficient check whether circles fit pairwise:

circle centre o₁ , radius r₁ in bottom left corner of triangle (marked angle α) , circle centre o₂ , radius r₂ in bottom right (marked b angle β)

the distance as₁ r₁/tan(½α), distance s₂b r₂/tan(½β), , distance s₁s₂ √((r₁ + r₂)² − (r₁ − r₂)²) = 2√r₁r₂

the circles fit if as₁ + s₁s₂ + s₂b ≤ ab.

in configuration (2) place 2 circles 2 of corners of triangle, , third circle between these 2 , 1 of 2 edges that's not touching both circles:

a thin triangle big green circle in top left corner, medium red circle in right corner, , blue circle between them , top edge

figuring out whether these fit bit more complex:

the thin triangle marked up: t altitude centre of big circle meets left edge, , s₁ s₃ altitudes centres of 3 circles meet bottom edge

to find length as₁ have walk round triangle corner c via point t. i'll leave details of exercise.

there eighteen ways arrange circles configuration.

is there configuration (3)? looked couldn't find 1 couldn't turned 1 of 2 gave. example, if 3 circles touch same side there's room swap middle circle on opposite side, getting configuration (2). however, enumeration of geometric configurations tricky , have missed one.


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