Hello, xoluosox!

An equilateral triangle has area: .

. . where is a side of the triangle.

Given a circle with radius , and an inscribed regular hexagon.

Prove that the area of hexagon is a mean proportional between the areas

of the inscribed and circumscribed equilateral triangles. Code:

* o *
* o o *
* r o o
* o o *
o o r
* o o *
* * *
* *
* *
* *
* *
* * *

The inscribed hexagon is composed of six equilateral triangles with side

Its area is: .

Code:

* o *
* /|\ *
* / | \ *
* / |r \ *
/ | \
* / | \ *
* / r * r \ *
* / * * \ *
/ * * \
o- - - - - - - - -o
* *
* *
* * *

It can be shown that the side of this triangle is: .

Then its area is: .

Code:

o
/|\
/ | \
/ | \
/ | \
/ | \
/ * * * \
/* | *\
* | *
* | o *
/ | o \
/* | o r *\
/ * * * \
/ * * \
/ \
/ * * \
/ * * \
/ * * \
o - - - - - - - * * *- - - - - - - - o

It can be shown that the side of this triangle is:

Its area is: .

If is the mean proportional between and

. . then: .

We have: .

. . which simplifies to: . . . . it's true!