Rational Trigonometry/7

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Solving Triangles with Rational Trigonometry

Case 1: A Right Triangle
This is the simplest case. The Pythagorean Theorem – $$Q_1 + Q_2 = Q_3$$, and the definitions of spread – $$s_1 = Q_1 / Q_3$$ and $$s_1 = Q_2 / Q_3$$ – are enough to solve everything. There are nice facts like $$s_1 + s_2 = 1$$ and the Spread Law.

Case 2: Non-Right Triangle
Typically, we know 3 of the six facts about a triangle (its three quadrances and its three spreads). If we know the three quadrances, we can use the Cross Law to find any particular spread. Alternatively, we could find the quadrea and thereby find any spread.

If we know two quadrances and a spread, then the Cross Law becomes a quadratic, with two solutions. Then we use the Spread Law to find the missing pieces.

Special Case: 1 Q, 2 S's
In the case of knowing one quadrance and two spreads, we must use the Triple Spread Law. It will give two solutions.

Lastly, if we know all three spreads, then the size could be anything.