List of Rational Trigonometry formulas

Pythagoras
If $$Q_1 + Q_2 = Q_3$$, the triangle is right.

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
$$(S_1+S_2+S_3)^2 = 2(S_1^2+S_2^2+S_3^2) + 4S_1S_2S_3$$

"Angle Chasing"
In other terms, a spread is equivalent to its supplement

Double
$$r = 4s(1-s) $$

Half
$$r = \frac{1 - sqrt{1-s}}{2}$$

Triple
$$r = s(3-4s)^2$$

Sum/Difference
$$A \pm B = A + B -2AB \pm 2\sqrt{A(1-A)B(1-B)}$$

Calculus
The rate of change of the spread at x increases is $$2\sqrt{x(1-x)}$$