Hello, furnis1!

First, make a sketch . . .

[I'll use Calculus to maximize OC.]

A semi-circle is dawn outwardly on a chord of a circle with center and raidus 1.

The point on this semi-circle is the furthest from center

Of course, the length of is dependant on the choice of chord

Determine so that is maximum.Code:* * * * * A * o * / :* o 1/ : o * / θ D: * * O * - - - + * - - o C * \ : * 1\ : o * \ :* o * o * * B * * *

The circle with center has radii

A semicircle is constructed (outward) with center and diameter

. . It has radii:

Let

In right triangle

Then: .

To maximize , set

We have: .

. . Hence: .

is maximum when

Then and is an isosceles right triangle.

Since its legs are length 1, its hypotenuse is: .