Circle-ception: The next level

This solution is similar to the one I posted for Circle-ception . This time, we bisect the 60° angle to form a right triangle with one leg $x$ and hypotenuse $10-x$ , where $x$ is the radius of the incircle.

Using the sine rule, $\frac { x }{ \sin { 30 } } =\frac { 10-x }{ \sin { 90 } } \\ \Rightarrow \frac { x }{ 0.5 } =\frac { 10-x }{ 1 } \\ \Rightarrow x=5-\frac { x }{ 2 } \\ \Rightarrow \frac { 3x }{ 2 } =5\\ \Rightarrow x=\frac { 10 }{ 3 } \\$

Since $x$ is the radius of the incircle, the diameter is $2x=\frac { 20 }{ 3 }$ , so $a=20$ and $b=3$ . $\left\lceil \frac { { 20 }^{ 2 }\times { 3 }^{ 2 } }{ 20+3 } \right\rceil =\left\lceil \frac { 3600 }{ 23 } \right\rceil =\left\lceil 156+\frac { 12 }{ 23 } \right\rceil =\boxed { 157 }$

isn't the brackets mean GIF?, if yes then it should be 156 , otherwise specify

Let A and B be the intersections of the circle with the sector that are not on the arc of the sector, C be the center of the circle, P be the center of the sector, and E be the intersection of the circle with the sector's arc. PC is a bisector of angle P, since C is equidistant from A and B. Thus the measure of angle CPB = 30 degrees. In addition, the angle PAC is a right angle, since PA is tangent to the circle. Thus we PAC is a 30-60-90 triangle, and PC = 2(AC). But PC = EC, so the diameter is 2/3 of PE, which is 10 since it is the radius of the circle. Thus a = 20, b = 3, and the value is the ceiling of 3600/23, which is 157.