The largest circle should be a circle tangent to both the radius of the sector and the radius of the arc, but it is a bit difficult to prove.

Let the center of the sector be A, the two radii are AB and AC, the center of the largest disk is O, AB is tangent to point D, AC is tangent to point E, and BC is tangent to point F.. (The symbols of some points are not necessary) Let the radius of the disk be r..

There is AO=OE√2=r√2.

AF=AO+OF=r√2+r= 10

So r =10/(√ 2+1) =10 (√ 2-1) (denominator is rational).

2.

Only 1

The bisector of each corner of a triangle intersects a point in a circle, and the distance from the point to the three sides is equal, so there are only 1 points, which are also the center of the inscribed circle of the triangle.