The center of a circle having radius 1 is denoted by O. The triangle ABC is inscribed within the circle such that the respective areas of the circular segments described by the sides AB, BC and AC are in the ratio 3:4:5.

A circle with its center located at P is inscribed within the triangle.

Determine the distance OP.

(In reply to Guesstimate by broll)

I used Mathematica to find a numerical solution.

For my approach I considered the angles formed from the circumcenter and the three sides, calling them x,y,z I used the constraints 0<x<2Pi/3, x<=y<=(2Pi-x)/2, z=2Pi-x-y to prevent checking duplicate triangles.  From this I calculated the areas of the 3 sections using area=2*Sin[angle/2] and then computed the sum of the squared errors from the required ratios of 3/4 and 4/5.  I then kept track of the lowest error value.  Initial search with step value of angles in radians of 0.01 gives a solution of
x=1.882, y=2.103, and z=2.298
which gives side lengths of
1.616, 1.736, and 1.825  (so broll, you seem good to 1 decimal place here)
also this gives the inradius of 0.49456
and this gives a distance between the two centers of
0.104

refinded search and got following to 5-digits accuracy
angles: 1.88245, 2.10269, 2.29805
sides: 1.61629, 1.73634, 1.82471
inradius: 0.49456
distance: 0.10431

Edited on September 20, 2010, 4:44 am

Posted by Daniel on 2010-09-20 02:44:23