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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 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
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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.
(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
- R. Nelsen, Proofs Without Words, MAA, 1993
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