Hello, sonia1!

A fascinating problem . . .

What is the largest figure of eight that will stand inside

an equilateral triangle of side length one?

(The circles are of the same size.) Code:

A
*
/|\
/ | \
/ * \
/* | *\
* r| *D
* *O *
/ * r| * \
/ * | * \
/ * \
/ * | * \
/ * r| * \
/ * * * \
/ * r| * \
/ * | * \
B * - - - - - - * - - - - - - - * C

We have equilateral triangle of side 1.

Two equal circles are "stacked" inside.

Let their radius be

Let be the center of the upper circle.

Draw .

Then

In right triangle ODA, we have: .

. . Then: .

Hence, the height of the triangle is

But the height of an equilateral triangle of side 1 is: .

Therefore: .