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) a (convex) equi-angular polygon
is constant.
(Note: while the above results can be extended to concave cases by using 'directed' distances and areas, it becomes tricky in some concave cases).
In the sketch below an equilateral pentagon is displayed, but navigate to an equi-angular pentagon, and a hexagon with opposite sides parallel using the Link to buttons in the sketch below.
Further generalizations of Viviani's Theorem
Challenge
1) Can you explain why (prove that) the results are true?
2) If you get stuck, try using the concept of a spiral similarity or 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.
Other examples involving equi-inclined lines
A generalization of Neuberg's Pedal Theorem & the Simson-Wallace line
Equi-inclined Lines Problem
Generalizations of a theorem by Wares
Equi-inclined Lines to the Sides of a Quadrilateral at its Vertices
The Equi-inclined Bisectors of a Cyclic Quadrilateral
The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral
Related Links
Distances in an Equilateral Triangle (Viviani's theorem) (Rethinking Proof activity)
2D Generalizations of Viviani's Theorem (Equilateral or equi-angled polygons or polygons with opposite sides parallel)
Miquel's Theorem (Rethinking Proof activity)
A variation of Miquel's theorem and its generalization
A generalization of Neuberg's Pedal Theorem to Polygons
Clough's Theorem (a variation of Viviani) and some Generalizations
External Links
Viviani's theorem (Wikipedia)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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