Here's how I would do that- it may not be suitable for you as it uses "coordinate geometry". Set up a coordinate system so that the origin is at the right angle, the positive y axis along the side of the triangle of length 9 and the positive x-axis along the side of length l. Let the center be at (a, b). We can immediately say that a= b since a is the distance from the center to the y axis, b is the distance from the center to the x axis, and those are both radii of the sphere. That is, the center of the sphere is at (a, a).
The hypotenuse runs from (12, 0) to (0, 9) and so has slope (9- 0)/(0- 12)= -9/12= -3/4. The equation of the hypotenuse is, in fact, or . A line perpendicular to that will have slope 4/3 and, in particular, a line perpendicular to the hypotenuse through (a, a) will have equation y= (4/3)(x- a)+ a= (4/3)x- (1/3)a. That line intersects the line when y= 9- (3/4)x= (4/3)x- (1/3)a. Solve for x and y to find point where the radius touches the hypotenuse, in terms of a, then calculate the distance between that point and (a, a), again in terms of a. Finally, set that distance equal to a, since we know the other two given radii have length a, and solve for a, the radius of the sphere.
By the way, this is only true if the sphere is completely in side the triangle- that is, that it is symmetric about the triangle. Since the sphere is three-dimensional and the triangle two-dimensional, you could have part of the sphere on one side of the triangle and a larger part on the other side. Then there would be an infinite number of possible solutions. In my opinion this problem would be better stated as a circle inside a triangle.