I found this collection of related problems by James Tanton on Twitter. Even though all these problems do not involve perpendiculars, they have a common solution approach – a sort of theme and variations idea. In a later tweet Tanton refers to a Viviani Theorem associated with these types of problems. I did not recall that theorem explicitly or by name. I also have not looked it up yet, in order to solve these problems on my own. I am guessing there is a more classical Euclidean geometry proof, but I like my vector approach for its clarity. I also throw in a bit a calculus at the end for fun. See Polygon Altitude Problems I