Rational Trigonometry/6

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Heron's Formula View Rationally

The typical formulation for the area of a triangle if one-half-base-times-height. This rests upon the formula for the area of a parallelogram: base-times-height.

If we use quadrances, we end up, not with the area, but with it squared. If we combine this with the Cross Law, we get Archimedes' Law

$$ 16\textrm{Area}^2 = 4Q_1Q_2-(Q_1+Q_2-Q_3)^2\\

= (Q_1+Q_2+Q_3)^2-2(Q_1^2+Q_2^2+Q_3^2) $$

Let is call this quantity the quadrea of the triangle (sixteen times the area squared). This is shown to be equivalent to Heron's Formula.