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 \(Q(A_1,A_2)=0. \blacksquare\)

(Midpoint If 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.