Equal Incircles along a Line

radius

1

number

4

Japanese Sangaku mathematics has a rich history of geometric theorems and problems. The equal incircles theorem is an illustrative example. Consider the incircle

C

of radius

r

of any triangle (each side of the triangle is tangent to the circle) and the extended line

L

along one side of the triangle. Construct a triangle with base common to

L

and an incircle of radius

r

on either side of the triangle. Continue this process on either side of the group of triangles until

n

triangles are constructed,

n>2

. The incircle of the triangle formed by the combination of any

a

,

2≤a<n

, adjacent triangles taken from the group of

n

individual triangles has the same radius as the incircle of the triangle formed by any other

a

adjacent triangles.