Viviani's Theorem: What is it?
A Mathematical Droodle
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).
Let Pa, Pb, and Pc be the pedal points (projections) of a point P on the side lines BC, AC, and AB of ΔABC. Then the theorem claims that
|(1)||h = PPa + PPb + PPc,|
where h is the length of the altitudes of ΔABC.
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:
|PPa + PPb + PPc||= TS + LU + LT|
|= VL + LU + TS|
One can arrive at the same conclusion without rotating the triangles:
The sides of the three small equilateral triangle add up to the side of the big one. Hence, the same holds for their altitudes.
- Ken-ichiroh Kawasaki, Proof Without Words: Viviani's Theorem, Mathematics Magazine, Vol. 78, No. 3 (June 2005), 213.