Varignon's theorem is a theorem of French mathematician
Pierre Varignon
Pierre Varignon (; 1654 – 23 December 1722) was a French mathematician. He was educated at the Society of Jesus, Jesuit College and the University of Caen, where he received his Magister Artium, M.A. in 1682. He took Holy Orders the following ...
(1654–1722), published in 1687 in his book ''Projet d'une nouvelle mécanique''. The theorem states that the
torque
In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...
of a
resultant
In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over th ...
of two concurrent forces about any point is equal to the
algebraic sum
In mathematics, summation is the addition of a sequence of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynom ...
of the torques of its components about the same point.
In other words, "If many
concurrent forces are acting on a body, then the algebraic sum of torques of all the forces about a point in the plane of the forces is equal to the torque of their resultant about the same point."
Proof
Consider a set of ''
''
force vector
In physics, a force is an influence that can cause an object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the magnitude and direction ...
s
that concur at a point
in space. Their resultant is:

:
.
The torque of each vector with respect to some other point
is
:
.
Adding up the torques and pulling out the common factor
, one sees that the result may be expressed solely in terms of
, and is in fact the torque of
with respect to the point
:
:
.
Proving the theorem, i.e. that the sum of torques about
is the same as the torque of the sum of the forces about the same point.
References
External links
Varignon's Theoremat
TheFreeDictionary.com
''The Free Dictionary'' is an American online dictionary and encyclopedia that aggregates information from various sources. It is accessible in fourteen languages.
History
The Free Dictionary was launched in 2005 by Farlex. In the same year, ...
Eponymous theorems of physics
Mechanics
Moment (physics)
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