Tangent bisectors

Geometry Level 5

In a triangle \(ABC\), \(BK\) is an angle bisector. A circle with radius \(\frac{5}{3}\) passes through the vertex \(B\), intersects \(AB\) at a point \(L,\) and is tangent to \(AC\) at \(K\). It is known that the length of \(AC\) is \(3\sqrt{3},\) and the ratio of the lengths \(|AK|\) to \(|BL|\) is \(6:5\). The area of the triangle \(ABC\) can be written as \( \frac{a\sqrt{b}}{c} \), where \(a\) and \(c\) are coprime positive integers, and \(b\) is not divisible by the square of any prime. What is the value of \(a+b+c\)?

Calvin Lin by Calvin Lin
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