Spread Law

The Spread Law states that for any triangle formed by $$A_1$$, $$A_2$$, and $$A_3$$ with with non-zero quadrances $$Q_1 = Q(A_2,A_3)$$, $$Q_2 = Q(A_1,A_3)$$, and $$Q_3 = Q(A_1,A_2)$$ and spreads $$ s_1 = s(A_1 A_2, A_1 A_3) $$, $$ s_2 = s(A_2 A_1, A_2 A_3) $$, and $$ s_3 = s(A_3 A_1, A_3 A_2) $$ follows the relationship:

$$ \frac{s_1}{Q_1} = \frac{s_2}{Q_2} = \frac{s_3}{Q_3} $$

Proof
If $$A_1$$, $$A_2$$, and $$A_3$$ are collinear then then the stament fallows from the the fact that $$ s_1 = s_2 = s_3 $$.

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 the Spread ratio implies

$$ s_2 = \frac{R_1}{Q_3} $$

$$ s_3 = \frac{R_1}{Q_2} $$

Solving for $$ R_1 $$ reveals

$$ R_1 = Q_3 s_2 = Q_2 s_3 $$

or

$$ \frac{s_3}{Q_3} = \frac{s_2}{Q_2} $$

in the same manner

$$ \frac{s_1}{Q_1} = \frac{s_2}{Q_2} $$