The sum of the
1) a (convex) equilateral polygon
2) a 2n-gon with opposite sides parallel
3) an equi-angular polygon
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
1) Can you explain why (prove that) the results are true?
2) If you get stuck, click
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.
Download an article of mine from the Mathematical Gazette (2002) giving a proof related to the results above from From nested Miquel triangles to Miquel distances.