Siegel identity

In mathematics, Siegel's identity refers to one of two formulae that are used in the resolution of Diophantine equations.

Statement
The first formula is


 * $$ \frac{x_3 - x_1}{x_2 - x_1} + \frac{x_2 - x_3}{x_2 - x_1} = 1 . $$

The second is


 * $$ \frac{x_3 - x_1}{x_2 - x_1} \cdot\frac{t - x_2}{t - x_3} + \frac{x_2 - x_3}{x_2 - x_1} \cdot \frac{t - x_1}{t - x_3} = 1 . $$

Application
The identities are used in translating Diophantine problems connected with integral points on hyperelliptic curves into S-unit equations.