Twelve Points On Two Orbits

Geometry Level 5

A triangle has vertices \(A=(0,0), B=(24,0), C=(6,12)\). The equation of its Steiner inellipse has the form: \[a{ y }^{ 2 }+bxy+{ cx }^{ 2 }+dy+ex+f=0\]

where \(a, b, c, d, e, f\) are co-prime integers with \(a>0\).

The equation of another ellipse that trisects all three sides of triangle \(ABC\) has the equation:

\[a{ y }^{ 2 }+bxy+{ cx }^{ 2 }+dy+ex+f_2=0\]

where \(a, b, c, d, e, f_2\) are co-prime integers, and \(a>0\).

If \(\dfrac{f}{f_2}=\dfrac{m}{n}\), where \(m,n\) are coprime positive integers, find \(m + n.\)

The problem is original

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