Rational Trigonometry/4

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Five Main Laws of Rational Trigonometry

Recall that quandrance is the square made with one side connecting two points. The spread between two lines is the opposite quadrance over the hypotenuse quadrance. Most people know Pythagoras's Theorem, that $$Q_1+Q_2=Q_3$$ on a right triangle, where $$Q_3$$ is the hypotenuse quadrance. Most people do not know the Triple Quad Formula, where $$(Q_1+Q_2-Q_3)^2=Q_1Q_2$$ precisely when $$Q_1$$ and $$Q_2$$ are collinear.

Spread Law
For any triangle, the spread divided by the quadrance is equal to any other spread divided by its quadrance.

Cross Law

 * $$(Q_1+Q_2-Q_3)^2 = 4Q_1Q_2(1-s_3)$$
 * $$(Q_2+Q_3-Q_1)^2 = 4Q_2Q_3(1-s_1)$$
 * $$(Q_3+Q_1-Q_2)^2 = 4Q_3Q_1(1-s_2)$$

Triple Spread Law
$$(s_1+s_2+s_3)^2=2(s_1^2+s_2^2+s_3^2) + 4s_1s_2s_3$$