Further generalizations of Viviani's Theorem

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

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.

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.

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Created 6 October 2011 by Michael de Villiers; modified 15 April 2019, 21 April 2020.