 Solution

Forthe chord-and-tangent group law gives, for distinct nonopposite points $P=(x_1,y_1)$ and $Q=(x_2,y_2)$,For doubling, replace the slope byThe point at infinity is the identity and $-(x,y)=(x,-y)$.

Let the $x$-coordinates of $P,Q,P+Q,P-Q$ be $x_1,x_2,x_3,x_4$. Applying the addition formula with $Q$ and $-Q$ givesSubstituting $y_i^2=x_i^3+a{x}_i+b$ and simplifying yieldsMultiplying the two addition formulas and eliminating $y_1{y}_2$ in the same way gives

These identities are the algebraic source of two parallelogram laws. Applied to pullbacks of the pole divisor of $x$, they implyfor isogenies of elliptic curves. Together with $\mathrm{deg}(n\phi )=n^2\mathrm{deg}\phi$, this makes the degree a quadratic form. Applied to the Absolute logarithmic Weil height of the four $x$-coordinates, with bounded terms removed by passage to the limit defining the canonical height of an elliptic curve, they similarly give