Let \triangle A B C be a triangle with side lengths a, b, and c. Circle \omega_{A} is the A-excircle of \triangle A B C, defined as the circle tangent to B C and to the extensions of A B and A C past B and C respectively. Let \mathcal{T}_{A} denote the triangle whose vertices are these three tangency points; denote \mathcal{T}_{B} and \mathcal{T}_{C} similarly. Suppose the areas of \mathcal{T}_{A}, \mathcal{T}_{B}, and \mathcal{T}_{C} are 4,5, and 6 respectively. Find the ratio a: b: c.