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Viviani's Theorem: What is it?
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Viviani's Theorem
The applet attempts to illustrate the following theorem:
| The sum of distances of a point inside an equilateral triangle or on one of its sides equals the length of its altitude. |
The theorem is named after Vincenzo Viviani (1622-1703).
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Let Pa, Pb, and Pc be the pedal points (projections) of a point P on the side lines BC, AC, and AB of 
ABC. Then the theorem claims that
| (1) | h = PPa + PPb + PPc, |
where h is the length of the altitudes of ABC.
To see that this is indeed so, draw the lines through P parallel to the sides of the triangle. I must mention that the current page has been prompted by a recent proof without words by Ken-ichiroh Kawasaki. The diagram below is just a slight modification of Kawasaki's proof:
Together with the side lines of the base triangle, the newly drawn lines form three equilateral triangles, which, between them, contain all the pieces in (1) as their altitudes. A one step rearrangement of the triangles followed by a rotation of two of them completes this delightful proof.
The applet above was originally intended to present a dynamic variant of Kawasaki's proof. It ended up as something a little different, hopefully still simple enough:
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(There is available another proof without words of Viviani's theorem. For you to compare. Also, the theorem extends to equilateral as well as equiangular polygons.)
References
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