# Geometry optimization

Three equilateral triangles with side lengths 28 are placed in the position as shown in the picture above. All the contacts are perfect and a circle passes by exactly one vertex per triangle. What’s the minimal radius?

• Does this have a neat, puzzle-like solution? – Dr Xorile May 27 at 22:11
• Yes, it does have one – Display maths May 27 at 22:30

$$84\left(\sqrt{3}-\sqrt{2}\right)\approx26.69832$$
I get this by assuming that the centre of the circle is $$x$$ along the line between the two joined triangles on the right, and that the apex of the triangle on the left is horizontal with this, which I think is correct. Solve for $$x$$ here. And substitute back to get the radius here.