# Find value of $\frac{\sum AA_1 \cos\left(\frac{A}{2}\right)}{\sum \sin A}$

Triangle $$\Delta ABC$$ is inscribed in a circle of radius one unit. If the internal angle bisectors of angles $$\angle A, \angle B,\angle C$$ meets the circle at the points $$A_1,B_1,C_1$$ respectively. Find value of $$S=\frac{\sum AA_1 \cos\left(\frac{A}{2}\right)}{\sum \sin A}$$

My try:

Letting $$BC=a, AB=c, AC=b$$

We have $$AD=\frac{2bc}{b+c}\cos\left(\frac{A}{2}\right)$$

Hence $$AA_1=AD+DA_1=\frac{2bc}{b+c}\cos\left(\frac{A}{2}\right)+DA_1$$

But how to find $$DA_1$$?