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!