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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