Triple quad formula proof
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This triple quad formula is a test for collinear points, one of the five basic laws of the rational trigonometry system devised in the early 2000s by Norman J. Wildberger.
It can either be proved by analytic geometry (the preferred means within rational trigonometry) or derived from Heron's formula, using the condition for collinearity that the triangle formed by the three points has zero area.
[edit] The formula
The three points are collinear precisely when
where Q(AB) is the "quadrance", i.e., the square of the distance, between A and B.
[edit] Proof by analytic geometry
The line has the general form:
where the (non-unique) parameters , and , can be expressed in terms of the coordinates of points and as:
so that, everywhere on the line:
But the line can also be specified by two simultaneous equations in a parameter , where at point and at point :
- and
or, in terms of the original parameters:
- and
If the point is collinear with points and , there exists some value of (for distinct points, not equal to 0 or 1), call it , for which these two equations are simultaneously satisfied at the coordinates of the point , such that:
- and
Now, the quadrances of the three line segments are given by the squared differences of their coordinates, which can be expressed in terms of :
where use was made of the fact that .
Substituting these quadrances into the equation to be proved:
Now, if and represent distinct points, such that is not zero, we may divide both sides by :
[edit] Derivation from Heron's formula
Three points are collinear if the triangle they enclose has zero area. By Heron's formula for area:
where s = the semiperimeter of the triangle,
Hence:
Substitute s, the semiperimeter, and multiply out:
Add 2a4 + 2b4 + 2c4 to both sides:
Factorise:
Quadrance being the square of length, this is equivalent to the triple quad formula:
Q.E.D.