# Playing With an Elliptic

Algebra Level 5

The equation of the elliptic curve shown here is

$16{ y }^{ 2 }={ x }^{ 3 }-25x.$

The thick blue line of the curve and the lines through points $$A, B, C, D$$ are all drawn accurately to scale. All four points have rational coordinates, but only one of them has an integer coordinate (either $$x$$ or $$y$$). The area of the triangle $$ABC$$ is rational and can be expressed as $$\frac { a }{ b }$$, where $$a$$ and $$b$$ are coprime integers.

Find the first 3 digits of the numerator $$a.$$

Lines through $$A$$ and $$D$$ are tangents to the elliptic curve, and both meet at point $$B$$ which is on the elliptic curve. Line through $$A$$ and $$D$$ intersect the elliptic curve at $$C$$.

Use of calculator or computer is recommended for computations.

Corrected courtesy of Jon Haussmann

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