# Finding the area of the region bounded by the incircle and the sides of the triangle?

In an isoscles triangle, we can find the radius of the incircle by using the fact that the angle bisector of the third (unequal) angle is the perpendicular bisector of the third(not equal) side and acts as a median and the other two angle bisectors are medians too. However, can we find the area of any one of the three sections of the triangle is divided into?( excluding the circle) ( please refer to the image)

Of course we can find the sum of the areas of the three sections by subtracting the area of the circle from the area of the triangle.

• Not sure how much you can use but I would find the equations of each line (take origin the be the lower left point for example) and find the area of each using an integral bounded by the equations of the lines and the circle. – user88595 Mar 1 '16 at 21:59

1.) From $BX = 0.5y$ and $IX = r$, find the area of $BXIY$ ($=m$, say).
2.). Find $\alpha$. From which, deduce $\beta$ and then $\delta$.
3.) Find the area of the blue sector ($=n$, say).
4.) Area of region $b = m + n - \pi r^2$