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 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 between the points and is the number
Sometimes will be called the quadrance from to or the quadrance of the side . Clearly
In rational or decimal number fields is always positive, and is zero precisely when . This is not necessarily the case of other fields.
In the complex number field with and
(Null line) If and are distant points, then is a null line precisely when
If and are distinct points, then by the Line through two points theorem
This is a null line precisely when which is exactly the condition that
(Midpoint If A1A2 is a non-null line, then there is a unique point lying on which satisfies This is the midpoint of the side . Furthermore
If and are distinct points, then by the Affine combination theorem(link required) any point on the line , has the form
Rewrite this as
By assumption is a non-null line, so by the previous Null line theorem, . Thus , and is the midpoint
of the side . Then
For , the numbers and are the qudarnces of the triangle, with the quadrance of the side , or the **quadrance opposite the vertex , and similarly for the other quadrances.