# Crazy Generalisation!

Geometry Level 4

Consider two circles $$\Gamma_{1}$$ and $$\Gamma_{2}$$ with centres $$O_{1}$$ and $$O_{2}$$ in the Euclidean plane. Let $$\Gamma_{1}$$ and $$\Gamma_{2}$$ have radii $$a$$ and $$b$$ units respectively, $$a > b > 0$$ and $$O_{1}O_{2}$$ be $$c$$ units, with $$c > (a+b)$$. Construct a direct common tangent $$T_{1}T_{2}$$ to these circles, with $$T_{1}$$ and $$T_{2}$$ on $$\Gamma_{1}$$ and $$\Gamma_{2}$$ respectively. Find $$\cos(\angle T_{1}O_{1}O_{2})$$ in terms of $$a,b$$ and $$c$$.

Note: The direct common tangent has both circles lie on the same side of the line.

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