# 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 $$m + n.$$

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