<p>"theres no other way at all to do it"</p>
<p>Actually, there is a way to do it that does not require to know the "obscure" formula. </p>
<p>As usual you need to pay attention to ALL the clues. One of them says that the radii are perpendicular. </p>
<p>Now, visualize the removal of the circle and draw three lines from the three angles to the center of the circle. This forms three smaller triangles with the radius as height. What do you have? The height of the three triangles is always the radius or 1. What it is the formula? bh/2. The sums of the three b is given as 16. This means that the sum of the 3 triangles is 16*1/2 or 8. </p>
<p>This said, I think this is a very hard problem, short of knowing the "obscure" formula or some knowledge of the incenter properties.</p>