Triangle Proportional Centers | Steve Wait – July 25, 2022

This exploration with specific regard to the prior posting of “Rhombi, Trapezia, and a Case of Pythagorean (et al.) Expansion“.

Alternative to the employment of lateral trapezia height to ascertain a triangle’s proportional center for use as a center of dilation, the following may be applied:

Where b lies along the x-axis and a intersects at x0y0 and within the isosceles limitation that occurs where c=b

$x=\left(\frac{ {b}}{2}\right)-\left(\frac{\left(\frac{\left(\frac{\left(Heron\right)}{.5ab}\right)\left({b}^n\left(\frac{.5 {ab}}{ {a}^n+ {b}^n+ {c}^n}\right)\right)}{.5b}\right)}{\left(\frac{a\times\cos{\left(C°-90°\right)}}{b+a\times\cos{\left(C°-90°\right)}}\right)}\right)$

$y=\frac{\left(\left(\frac{ {Heron}}{.5ab}\right)\left({b}^n\left(\frac{.5 {ab}}{ {a}^n+ {b}^n+ {c}^n}\right)\right)\right)}{.5b}$

Interesting is a trace of the proportional center’s path for a given a:b ratio. As example, an a:b of .75 would likely appear as follows (note the extending of the path beyond the isosceles inversion):