Question 17
In triangle $ABC$, $\angle A=60^\circ$ and the lengths $AB,AC$ are the roots of $x^2-5x+3=0$. The point $D$ is on $BC$ and $AD$ bisects $\angle A$. The lines $AB,AC$ are tangent to a circle with center $D$. Its radius is $\dfrac{a\sqrt b}{c}$, where $a,c$ are coprime positive integers and $b$ is a positive integer with no square factor greater than 1. Find $a+b+c$.
Tangency is to the lines AB and AC, extended beyond the side endpoints if necessary, as in the source diagram.
Tangency is to the lines AB and AC, extended beyond the side endpoints if necessary, as in the source diagram.