A scalene triangle has three sides of different lengths. Likewise, it has three different heights depending on which side is base. In designing the Freedom Dobro Capo, I needed to calculate the coordinates of a scalene-triangle's apices from its three heights.
The formulae are given below the accompanying figure, where h1 is height when the triangle is oriented with b1 as base (as in the example), h2 is height when b2 is base, and h3 is height when b3 is base. The triangle is positioned with its left-base apex at the origin (where both x and y coordinates are 0) and with its b1 on the x axis. The x coordinate of the right-base apex has the value of b1. The x coordinate of the top apex is x1. The y coordinates of these apices are 0 and h1 respectively. The formulae give b1 and x1 from specified h1, h2 and h3.
They are derived from the formula for area of a triangle and the pythagorean theorem.
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