At two places in the proof, varignon refers to part. These forces are represented in magnitude and direction by. Telling students to construct the diagonals of the parallelogram feels like were depriving them of the aha. Varignons theorem need not be restricted to the case of two components, but it applies equally well to three or more. Varignons theorem states that the moment at any point of the force acting on a body is equal to. Coxeter and greitzer attribute the theorem to pierre varignon 16541722. Varignons theorem is a statement in euclidean geometry, that deals with the construction of a particular parallelogram, the varignon parallelogram, from an arbitrary quadrilateral quadrangle. The area of the varignon parallelogram of a convex quadrilateral is half that of the quadrilateral, and the perimeter is equal to the sum of the diagonals of the original quadrilateral. The forces p and q represent any two nonrectangular components of r.
We are currently experiencing intermittent display of pdf links on some article pages. I have a problem with varignons theorem the midpoints of the sides of an arbitrary quadrangle form a parallelogram. The answers are given in the next theorem and its proof. Their resultant r is represented in magnitude and direction by oc which is the diagonal of parallelogram oacb. In other words, if many concurrent forces are acting on a body, then. Thus we could have used any number of concurrent components of r in the foregoing proof. This theorem was first formulated and proved by the french scientist p.
The principle of moments or varignons principle is not restricted to only two concurrent forces but is also applicable to any coplanar force system, i. Fig 33 moment a m using varignons principle to illustrate. Varignons theorem is a theorem by french mathematician pierre varignon 16541722, published in 1687 in his book projet dune nouvelle mecanique. I did this putting the d point in the perpendicular of ac through b. Physics assignment help, varignon s theorem, proof of varignon s theorem. Determine a the total moment at the base of the structure. Varignons theorem mathematical association of america. I think items above showcase dynamic geometry at its best, but im less enthusiastic about dynamic geometry softwares role in the proof of varignons theorem in item 4.
This parallelogram, known as the varignon parallelogram, has an area equal to half the area of the original quadrilateral, and a perimeter equal to the sum of the lengths of the original quadrilaterals diagonals. It is named after pierre varignon, whose proof was published posthumously in 1731. Quadrilateral, midpoints, parallelogram, area, perimeter. Varignons theorem the figures for the article varignons theorem for octahedra and cross polytopes by john pesek that appeared in the june, 2009 issue of mathematics magazine are displayed here. Varignons theorem states that the sum of the moments produced by any two concurrent forces with respect to a point is equal to the moment produced by the resultant with respect to the same point. To prove varignons theorem, consider the force r acting in the plane of the body as shown in the aboveleft side figure a. With the help of vector notation of moment the resultant of forces p and q is r as shown in fig. The theorem states that the torque of a resultant of two concurrent forces about any point is equal to the algebraic sum of the torques of its components about the same point. It essentially states that the moment of a force about a pivot point is equal to the sum of the moments of the components of that force about the point.
In the above right side figure b illustrates the usefulness of varignons theorem. Nolans proof of varignons theorem free download as pdf file. Leibniz, furnished the first rigorous proof that a parallelogram is formed. It states if number of coplanar forces is acting simultaneously on a particle, algebraic sum of moments of all the forces about any point is equal to moment of their. If the diagonals of the quadrangle are perpendicular then the parallelogram is a rectangle. Varignons theorem states that the moment of a force about any point is equal to the algebraic sum of the moments of its components about that. For professor akbar of ut dallas for geometry 3321.
To illustrate varignons principle by an example example 3. Elements of civil engineering and mechanics 1 answer prove that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. These forces are represented in magnitude and direction by oa and ob. Varignons theorem if two or more coplanar force is acting on a body then sum of moments of these forces about any point in the plane of. The midpoint of the segment joining the points x 1, y 1 and x 2, y 2 is the point x 1. In this paper i illustrate how learners can use geogebra to discover, visualize, and investigate variations of varignons theorem. The area of varignons parallelogram is one half that of.
If you are unable to see the pdf link please follow the. Proving varignons theorem plus a little history behind the man himself. Extending varignons theorem to nonstandard quadrilaterals 4. Varignons theorem ii the area of the varignon parallelogram is half that of the corresponding quadrilateral, and the perimeter of the parallelogram is equal to the sum of the diagonals of the original quadrilateral. The figure formed when the midpoints of the sides of a convex quadrilateral are joined in order is a parallelogram. Investigating variations of varignons theorem using geogebra jose n. Investigating variations of varignons theorem using geogebra. Varignons theorem, proof of varignons theorem, physics. Varignons theorem given a mass force w 200 kn is supported by one cable on the indicated structure. The figure formed when the midpoints of the sides of a quadrangle are joined is a parallelogram pierre varignon, 1731.
Fig a shows two forces fj and f2 acting at point o. In varignons arrangement of the euclidean geometric propositions, part. Digging deep into varignons theorem sine of the times. The application of varignons theorem practically is to find the position of the resultant from any point of the body. Given its simplicity, it is utterly surprising that it had to be waiting that long to be discovered. Varignons theorem and its proof hindi engineering mechanics. According to varignons theorem, if a system of forces f i has a resultant force r, then the moment m 0 r of the resultant force relative to any center o or zaxis is equal to the sum of the moments m 0 f i of the component forces relative to the same. Varignons theorem of moments resolution and composition of.
Varignons theorem is one of my favorite results in elementary geometry. What links here related changes upload file special pages permanent link. I hope you looked up varignons theorem as it applies to moments. Varignons theorem mechanical engineering assignment. The moment of the resultant of a number of forces about any point is equal to the algebraic sum of the moments of all the forces about the same point. Varignon theorem mechanics, mechanical engineering. Varignons theorem, introduction to varignons theorem youtube.1610 1521 1596 178 946 130 1490 888 130 605 1569 899 1486 497 951 1439 1388 699 1556 951 1563 603 790 1269 1410 880 447 952 1256 914 1197 544 556 715 924 90 425