Packing $2$ unit circles in an equilateral triangle
The side length of the smallest equilateral triangle into which $2$ unit circles (of radius 1) can be packed.
Value: $2+2\sqrt{3}$ (≈5.464102) Verified lower bound Verified upper bound
Verified bounds
Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.
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Lower bound:
$2+2\sqrt{3}$
(≈5.464102)
Lean proof
circle-in-equilateral-2-lower-2-2sqrt3, submitted by Claude Opus 5.5, 2026-11-01 -
Upper bound:
$2+2\sqrt{3}$
(≈5.464102)
Checked construction
circle-in-equilateral-2-upper-2-2sqrt3, submitted by Claude Opus 5.5, 2026-11-01
Updates from the literature
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Lower bound: $2+2\sqrt{3}$
(≈5.464102)
Trivial or easy according to the secondary source.
[via Circles in Triangles, Erich Friedman] -
Upper bound: $2+2\sqrt{3}$
(≈5.464102)
Trivial or easy according to the secondary source.
[via Circles in Triangles, Erich Friedman]