# Quadrance

**For the beginner: Quadrance is just distance squared but for the advanced this is not the appropriate notion.**

This section begins with the concept of the $quadrance$ between points, and then examines null lines, midpoints, the Triple Quad Formula, Pythegoras' theorem, the quadrea of a triangle, and the generalization of the classical Heron's formula called -- in this wiki--Archimedes' formula. Perpendicular bisectors and quadrance from a point to a line are discussed. Archimedes' function and the Quadruple quad formula are defined. All these topics hold in an arbitrary field F not of characteristic two (from now on this will not necessarily be repeated).

The

quadrance$Q(A_1,A_2)$between the points $A_1\equiv [x_1,y_1]$ and $A_2 \equiv [x_1,y_2]$ is the number $Q(A_1,A_2) \equiv (x_2-x_1)^2+(y_2-y_1)^2$

Sometimes $Q(A_1,A_2)$ will be called the quadrance **from $A_1$to$A_2$** or the **quadrance of the side $\overline{A_1A_2}$**. Clearly $Q(A_1,A_2)=Q(A_2,A_1).$

In rational or decimal number fields $Q(A_1,A_2)$ is always positive, and is zero precisely when $A_1=A_2$. This is not necessarily the case of other fields.

In the complex number field with $A_2 \equiv [0,0]$ and $A_2 \equiv [1,i]$

$Q(A_1,A_2) = 1^2 +i^2=0. \diamond$

(Null line)If $A_1$ and $A_2$ are distant points, then $A_1A_2$ is a null line precisely when $Q(A_1,A_2)=0.$

If $A_1\equiv [x_1,y_1]$ and $A_2\equiv [x_2,y_2]$ are distinct points, then by the Line through two points theorem

$A_1 A_2 = \langle y_1-y_2: x_2-x_1:x_1y_2-x_1y_2-x_2y_1\rangle$

This is a null line precisely when $(y_1-y_2)^2 + (x_2-x_1)^2=0$ which is exactly the condition that $Q(A_1,A_2)=0. \blacksquare$

(MidpointIf A1A2 is a non-null line, then there is a unique point $A$ lying on $A_1A_2$ which satisfies $Q(A_1,A) = Q(A,A_2).$ This is the midpoint $M\equiv \frac{1}{2}A_1 + \frac{1}{2}A_2$ of the side$\overline{A_1A_2}$.Furthermore $Q(A_1,M) = Q(M,A_2) = \frac{Q(A_1,A_2)}{4}.$

If $A_1\equiv [x_1,y_1]$ and$A_2 \equiv [x_2,y_2]$ are distinct points, then by the Affine combination theorem(link required) any point on the line $A_1A_2$, has the form

$A \equiv \lambda A_1 +(1-\lambda)A_2 = [\lambda x_1 + (1-\lambda) x_2, \lambda y_1 + (1-\lambda) y_2 )^2$ $= (\lambda x_1 +(-\lambda)x_2)^2 + (\lambda y_1 +(-\lambda)y_2)^2.$ .

Rewrite this as $\left( (1-\lambda)^2 -\lambda^2 \right) \left( (x_2 - x_1)^2 + (y_2 - y_1)^2 \right) = 0$

By assumption $A_1A_2$ is a non-null line, so by the previous Null line theorem, $x_2-x_1)^2 +(y_2-y_1)^2 \not = 0$. Thus $\lambda = \frac{1}{2}$, and $A$ is the midpoint

$M \equiv \frac{1}{2} A_1 + \frac{1}{2}A_2 = \left[ \frac{(x_1 +x_2)}{2}, \frac{(y_1 +y_2)}{2} \right]$

of the side $\overline{A_1A_2}$. Then

$Q(A_1,M)= \left( \frac{x_1-x_2}{2} \right) + \left( \frac{y_1-y_2}{2} \right) = Q(M,A_2)$ $= \frac{Q(A_1,A_2)}{4} \blacksquare$

For $\triangle A_1A_2A_3$ , the numbers $Q_1 \equiv Q(A_2,A_3), Q_2\equiv Q(A_1,A_3)$ and $Q_3 \equiv Q(A_1,A_2)$ are the qudarnces of the triangle, with $Q_1$ the

quadrance of the side $\overline{A_2A_3}$, or the **quadrance opposite the vertex$\overline{l_1l_2}$, and similarly for the other quadrances.