Playing With an Elliptic

Algebra Level 5

Corrected courtesy of Jon Haussmann

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.


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