Hello, amerlaw1!

2. What is the distance between the incenter and circumcenter

of a triangle with sides of the length 12, 16 and 20? Code:

_ A *
: | *
: | *
: | *
: | *
: | *
: | * 20
: | * * * *
: | * * E
12 | * o
: |* r * *
: | * *
: * * * *
: F o - - - - o * *
: * r |O * *
: | | *
: r|* |r * *
: | * | * *
: | * | * *
- C * - - - * o * - - - - - - - - - - - - * B
r D
: - - - - - - - - 16 - - - - - - - - :

We have right triangle

Place the triangle on a coordinate system with at the Origin.

The incenter is

. .

The circumcenter is the midpoint of

. . (Not shown on the diagram.)

Note that: .

Since is also tangent to the circle:

Note that: .

Since is also tangent to the circle:

Since , we have: .

We have: .

Therefore: .