Quadrance

For those from common mathematics backgrounds, quadrance appears to be the square of the distance between two points. However, in Rational Trigonometry, it is seen to be more fundemental than distance. In Analytic Geometry, it is sum of the square of the difference of x-coordinates with the square of the difference of the y-coordinates. Formally, the quadrance $$ Q(A_1,A_2) $$ between points $$A_1 \equiv [x_1,y_1]$$ and $$A_2 \equiv [x_2,y_2]$$ is the number

$$Q(A_1,A_2) \equiv (x_2-x_1)^2 + (y_2 - y_1) ^ 2$$

In fields other than the Rationals, it is possible to have a quadrance of 0 for non-identical points, e.g. $$(0,0)$$ and $$(1,i)$$. In terms of vectors, the quadrance is the same as the dot-product.

Because quadrance is inherently a quadratic quantity, the number 4 appears quite often in formulas and equations. For example, the quadrance to the midpoint of two points has 1/4 the quadrance of the overall segment.

Colinear points follow this formula:

$$(Q_1+Q_2+Q_3)^2 = 2(Q_1^2+Q_2^2+Q_3^2)$$

Add the quadrances of two line segments is a quadratic problem.

A triangle relates the sides and one spread

$$ (Q_3-Q_2-Q_1)^2 = 4Q_1Q_2(1-S_3) $$