# Problem 1. The optimal triangle

- Author:
- Alla Stolyarevska

A circle of radius R lies on a plane. The two circles with radii R1 and R2, such that R1 + R2 = R, are cut out of the circle. Find a triangle of largest area that can be inscribed in a figure obtained after cutting.

**The student’s decision.**On the sheet we can see the structure composed of the circles of radii R = FB = 3, R1 = OA = 1 and R2 = OD = 2. The circles of radii R1, R2 have a common tangent

*a*. G and H are the points of tangency. The secant line to a circle of radius R is drawn through the points G and H. The points J, K of secant's intersection with the circle are connected with the point N, which is the point of intersection of the perpendicular drawn from the middle of the segment JK to the circle. JKN is the required triangle.