Cross Law

The Cross law states that three points $$A_1$$, $$A_2$$, and $$A_3$$ with quadrances $$Q_1 = Q(A_2,A_3)$$, $$Q_2 = Q(A_1,A_3)$$, and $$Q_3 = Q(A_1,A_2)$$ and spread $$ s_3 = s(A_3 A_1, A_3 A_2) $$ fallow the relationship:

$$ (Q_1 + Q_2 - Q_3)^2 = 4Q_1 Q_2 (1 - s_3) $$

Proof
If $$A_1$$, $$A_2$$, and $$A_3$$ are colinear then s_3 = 0 then the stament fallows from the Triple Quad Formula.

Otherwise the points form a triangle so let D be the foot of the altitude from $$ A_1 $$ to the line $$ A_2 A_3 $$ and define the quadrances:

$$ R_1 = Q(A_1,D) R_2 = Q(A_2,D) R_3 = Q(A_3,D) $$

Becaues $$ A_2 A_3 $$ and $$ A_1 D $$ are perpendicular Pythagoras' Theorem implies

$$ Q_3 = R_1 + R_2 $$

$$ Q_2 = R_1 + R_3 $$

so by the spread ratio $$ 1 - s_3 = \frac{R_3}{Q_2} $$ Solving for $$R_1$$, $$R_2$$, and $$R_3$$ yeilds:

$$ R_3 = Q_2 (1 - s_3) $$

$$ R_1 = Q_2 s_3 $$

$$ R_2 = Q_3 - Q_2 s_3 $$

$$Q_1$$, $$R_2$$, and $$R_3$$ are collinear so the Triple Quad Formula implies

$$ (Q_1 + R_2 - R_2)^2 = 4Q_1 R_3 $$

which by substitution is:

$$ (Q_1 + Q_2 - Q_3)^2 = 4Q_1 Q_2 (1 - s_3) $$