Viviani's Theorem: What is it?
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Copyright © 1996-2012 Alexander BogomolnyViviani'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 ΔABC. Then the theorem claims that
| (1) | h = PPa + PPb + PPc, |
where h is the length of the altitudes of ΔABC.
Consider a shifted copy A'B'C' of ΔABC such that P lies on B'C'. Let PS be parallel to AB, Q be the foot of the perpendicular from C' onto PS, and L the foot of the perpendicular from P to A'C', as in the applet. Then
| h | = C'Q + PPc |
| = PL + PPc | |
| = PbL + PPb + PPc | |
| = PPa + PPb + PPc. |
A slightly incorrect version of the above appears as a proof without words in [Nelsen, p. 15].
A direct proof is also pretty simple. As long as point P is not outside ΔABC, we have
| (2) | Area(ABC) = Area(ABP) + Area(BCP) + Area(CAP). |
Let m be the length of a side of ΔABC. Then from (2)
m·h/2 = PPc· m/2 + PPa· m/2 + PPb· m/2,
wherefrom (1) follows immediately.
Note
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.
There is also a very simple proof by inversion which reduces Viviani's theorem to the identity
AB + BC = AC, where A, B, C, are three collinear points with B between A and C.Finally, Viviani's theorem follows from a property of isosceles triangles.
References
- R. Nelsen, Proofs Without Words, MAA, 1993
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