geometry

I would draw this in the first quadrant on a sheet of graph paper.

I would place the right angle (C) at the origin.

I would have the "4" side on the horizontal axis, with the point B being at (4,0).

I would have the "3" side on the vertical axis, with the point A being (0,3).

 

The circumcenter of a right triangle is the midpoint of the hypotenuse.

The coordinates of the midpoint of AB are (2, 1.5).

 

To find the incenter (which is found at the point of intersection of the angle bisectors:

-- the angle bisector of angle(C) is on the line y = x. 

-- to find the angle bisector of B, I'll find the size of angle(B), divide that in two, and find its tangent.

     -- to find the size of angle(B), find the tangent(B)  =  3/4

     -- turn that into degrees.

    -- divide those degrees by 2

    -- find the tangent of that angle; which is 1/3.

-- use this slope to count back from B three steps and up one step, giving you the point (1, 1) 

     -- this point is on the line y = x, so this point is the incenter.

 

Circumcenter  =  (2, 1.5)

Incenter  =  (1, 1)

 

Use the distance formula ...