The sum of the Miquel distances (obtained by drawing equi-inclined (Miquel) lines with fixed angles to the sides) from a point P to the sides of: 1) a (convex) equilateral polygon
2) a 2n-gon with opposite sides parallel
3) an equi-angular polygon is constant.

In the sketch below an equilateral pentagon is displayed, but navigate to a hexagon with opposite sides parallel, and an equi-angular pentagon using the Link buttons below.

Further generalizations of Viviani's Theorem

Challenge
1) Can you explain why (prove that) the results are true?
2) If you get stuck, click here for a Hint.

Note that the results hold even when the feet of the Miquel lines fall on the extensions of the sides or P is moved outside the polygon provided we regard distances falling completely outside the polygon as negative; in other words using directed distances (or vectors). However, WebSketchpad does not measure 'negative' distances, so dragging P outside will appear to no longer give a constant sum.