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=(b2)−((((Heron).5ab)(bn(.5aban+bn+cn)).5b)(a×cos(C°−90°)b+a×cos(C°−90°)))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=((Heron.5ab)(bn(.5aban+bn+cn))).5by=\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,...
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