Hello, Lukybear!

The bottom row looks like this:

Code:

/ \
* * * / \ * * *
* /* *\ *
* / * * \ *
* / * * \ *
/ \
* / * \ *
* o - - - - * - - - - o - - - - *
* * | r * r | *
* | |
* |r * * |r *
* * | * * | *
* * | * * | *
A o---------------*-*-*---------------*-*-*--------------
: - - - r√3 - - - : - - - 2r - - - - :

is a vertex of the triangular frame.

There are four circles across the bottom row.

We can see that the bottom of the frame is:

. .

Hence: .

. . which rationalizes to: .

(b) Show that the outside perimeter of the figure which remains

when the frame is removed is: .

The perimeter of each circle is: .

There are three circles at the vertices of the triangle.

. . Each has an exposed perimeter of: .

The three "corner circles" have a combined perimeter of

There are six circles at the sides of the triangle.

. . Each has an exposed perimeter of: .

The six "side circles" have a combined perimeter of

The total perimeter of the figure is: .

Since

. .