In the following geometry figure with a circle and triangle, find the measure of the largest side length As title suggests, the question is to find the largest length's measure (the blue length) in the figure below. Note that the blue length is NOT passing through the center of the circle and the orange length is tangent to the circle. I tried several different approaches to this problem, both purely geometric and trigonometric, but none of them provided a simple solution until I recalled an important theorem. Please post your own solutions as well, I will have mine as an answer below!

 A: My solution is pretty forward:

We can label the triangle as $\triangle ABC$ with $D$ on $BC$. Since we know that $AB$ is tangent to the circle, we can use the Alternate Segment Theorem and say that $\angle BAD=\angle BCA=\alpha$. Next we can label $\angle ABC=\beta$.
Now, notice that $\triangle BAD$ and $\triangle BCA$ are similar via the AAA property, thus the relation:
$$\frac{2}{4}=\frac{3}{BC}$$
$$BC=6$$.
A: Triangle cosine Rule applied twice, alternate segment angle, outside tangent squared is product segment theorem are used
I quickly got two  results by above trig numerical method. It is only slightly different from the first one by @Goku. Others complex, discarded. Can you spot my mistake in the second solution on Mathematica ? Or, is a second solution also admissible?
Two values of long side $ BC= x+y ~$ are
$$(1.5+4.5, 1.50735+4.46341=6, 5.97076)$$

$ (x,y)$ are segments in the blue line outside  and inside respectively.
$$ ( x^2=3^2+2^2-2\cdot 3 \cdot 2~\cos \beta,~y^2+4^2-2 \cdot 4 y ~\cos \beta = 2^2,~ x*(x+y)=9 ~)$$

