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.

Daniel:

Using an iterative method ( theta(area1)+theta(area2)+theta(area3) = 2pi), I have now obtained:

area(medium) approx. 0.620418 (and less than 0.62041801)

This gives:

theta (small) = 1.8824525260936  
theta (med) = 2.10268483077685  
theta (big) =  2.2980473161196 

Then using   theta = 2 arcsin(c/[2r])

Chord(small) 1.61656145662681292
Chord(med) 1.73618078168484268
Chord(big) 1.8247293637203118

Then abc/2(a+b+c) = 0.494581541698617

Using d^2 = R(R-2r)

 d = 0.10410051201971

I note with simply enormous relief that your findings are not that different!

 

 

Edited on September 20, 2010, 3:45 am

Posted by broll on 2010-09-20 03:40:33