Rational Trigonometry/2

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Quadrance via Pythagoras and Archimedes
Most of Ancient Greece was concerned with points, lines, and circles. Important concepts included parallelism, perpendicularity, and congruence. Quadrance was defined as an area made out of a line-segment, turned into a square. This was how equivalence was established. The two main results of this were: the Pythagorean Theorem and the Triple Quad Formula. Of these, one is known everywhere, and the other is almost unknown.

The Triple Quad Formula can be stated as

$$4Q_1Q_2 = (Q_1 + Q_2 - Q_3)^2$$,

or in its symmetric form

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

This leads immediately to Archimedes's Theorem, because if the two sides are equal, they mark out a triangle of size zero. If the two quadrances are not laid out colinearly, however, they describe a triangle:

$$16A^2 = 4Q_1Q_2-(Q_1+Q_2-Q_3)^2 = (Q_1+Q_2+Q_3)^2-2(Q_1^2+Q_2^2+Q_3^2)$$