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Varignon's theorem
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<h2 id="theorem">Theorem</h2> <p>The midpoints of the sides of an arbitrary quadrilateral form a parallelogram. If the quadrilateral is <a href="/facts/Convex_polygon/7x35w610">convex</a> or <a href="/facts/Concave_polygon/opSnBwgh">concave</a> (not <a href="/facts/Quadrilateral/QCGqrkp2">complex</a>), then the <a href="/facts/Area/KsNG1G9t">area</a> of the parallelogram is half the area of the quadrilateral. </p><p>If one introduces the concept of oriented areas for <a href="/facts/Polygon/yAq7RjPz"><i>n</i>-gons</a>, then this area equality also holds for complex quadrilaterals.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>The Varignon parallelogram exists even for a <a href="/facts/Quadrilateral/QCGqrkp2">skew quadrilateral</a>, and is planar whether the quadrilateral is planar or not. The theorem can be generalized to the <a href="/facts/Midpoint_polygon/2TjI19kW">midpoint polygon</a> of an arbitrary polygon. </p> <h2 id="proof">Proof</h2> <p>Referring to the diagram above, <a href="/facts/Triangle/cRXUwsMn">triangles</a> <i>ADC</i> and <i>HDG</i> are similar by the side-angle-side criterion, so <a href="/facts/Angle/gFUTEQYq">angles</a> <i>DAC</i> and <i>DHG</i> are equal, making <i>HG</i> parallel to <i>AC</i>. In the same way <i>EF</i> is parallel to <i>AC</i>, so <i>HG</i> and <i>EF</i> are parallel to each other; the same holds for <i>HE</i> and <i>GF</i>. </p><p>Varignon's theorem can also be proved as a theorem of affine geometry organized as linear algebra with the linear combinations restricted to coefficients summing to 1, also called affine or <a href="/facts/Barycentric_coordinates_(mathematics)/ymlRaudU">barycentric coordinates</a>. The proof applies even to skew quadrilaterals in spaces of any dimension. </p><p>Any three points <i>E</i>, <i>F</i>, <i>G</i> are completed to a parallelogram (lying in the plane containing <i>E</i>, <i>F</i>, and <i>G</i>) by taking its fourth vertex to be <i>E</i> − <i>F</i> + <i>G</i>. In the construction of the Varignon parallelogram this is the point (<i>A</i> + <i>B</i>)/2 − (<i>B</i> + <i>C</i>)/2 + (<i>C</i> + <i>D</i>)/2 = (<i>A</i> + <i>D</i>)/2. But this is the point <i>H</i> in the figure, whence <i>EFGH</i> forms a parallelogram. </p><p>In short, the <a href="/facts/Centroid/H6dDbg0h">centroid</a> of the four points <i>A</i>, <i>B</i>, <i>C</i>, <i>D</i> is the midpoint of each of the two diagonals <i>EG</i> and <i>FH</i> of <i>EFGH</i>, showing that the midpoints coincide. </p><p>From the first proof, one can see that the sum of the diagonals is equal to the perimeter of the parallelogram formed. Also, we can use vectors 1/2 the length of each side to first determine the area of the quadrilateral, and then to find areas of the four triangles divided by each side of the inner parallelogram. </p> <table><tbody><tr><th>convex quadrilateral</th><th>concave quadrilateral</th><th>crossed quadrilateral</th></tr><tr><td></td><td></td><td></td></tr></tbody></table> <h2 id="the-varignon-parallelogram">The Varignon parallelogram</h2> <h3>Properties</h3> <p>A planar Varignon parallelogram also has the following properties: </p> <ul><li>Each pair of opposite sides of the Varignon parallelogram are parallel to a diagonal in the original quadrilateral.</li> <li>A side of the Varignon parallelogram is half as long as the diagonal in the original quadrilateral it is parallel to.</li> <li>The area of the Varignon parallelogram equals half the area of the original quadrilateral. This is true in convex, concave and crossed quadrilaterals provided the area of the latter is defined to be the difference of the areas of the two triangles it is composed of.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li> <li>The <a href="/facts/Perimeter/yQv1AajA">perimeter</a> of the Varignon parallelogram equals the sum of the diagonals of the original quadrilateral.</li> <li>The diagonals of the Varignon parallelogram are the bimedians of the original quadrilateral.</li> <li>The two bimedians in a quadrilateral and the line segment joining the midpoints of the diagonals in that quadrilateral are <a href="/facts/Concurrent_lines/zPT6S4w9">concurrent</a> and are all bisected by their point of intersection.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>: p.125 </li></ul> <p>In a convex quadrilateral with sides <i>a</i>, <i>b</i>, <i>c</i> and <i>d</i>, the length of the bimedian that connects the midpoints of the sides <i>a</i> and <i>c</i> is </p> m = 1 2 − a 2 + b 2 − c 2 + d 2 + p 2 + q 2 {\displaystyle m={\tfrac {1}{2}}{\sqrt {-a^{2}+b^{2}-c^{2}+d^{2}+p^{2}+q^{2}}}} <p>where <i>p</i> and <i>q</i> are the length of the diagonals.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> The length of the bimedian that connects the midpoints of the sides <i>b</i> and <i>d</i> is </p> n = 1 2 a 2 − b 2 + c 2 − d 2 + p 2 + q 2 . {\displaystyle n={\tfrac {1}{2}}{\sqrt {a^{2}-b^{2}+c^{2}-d^{2}+p^{2}+q^{2}}}.} <p>Hence<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>: p.126 </p> p 2 + q 2 = 2 ( m 2 + n 2 ) . {\displaystyle \displaystyle p^{2}+q^{2}=2(m^{2}+n^{2}).} <p>This is also a <a href="/facts/Corollary/ntQ8zR9N">corollary</a> to the <a href="/facts/Parallelogram_law/JI1LwPxd">parallelogram law</a> applied in the Varignon parallelogram. </p><p>The lengths of the bimedians can also be expressed in terms of two opposite sides and the distance <i>x</i> between the midpoints of the diagonals. This is possible when using Euler's quadrilateral theorem in the above formulas. Whence<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> m = 1 2 2 ( b 2 + d 2 ) − 4 x 2 {\displaystyle m={\tfrac {1}{2}}{\sqrt {2(b^{2}+d^{2})-4x^{2}}}} <p>and </p> n = 1 2 2 ( a 2 + c 2 ) − 4 x 2 . {\displaystyle n={\tfrac {1}{2}}{\sqrt {2(a^{2}+c^{2})-4x^{2}}}.} <p>The two opposite sides in these formulas are not the two that the bimedian connects. </p><p>In a convex quadrilateral, there is the following <a href="/facts/Duality_(mathematics)/8jGuyQIU">dual</a> connection between the bimedians and the diagonals:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <ul><li>The two bimedians have equal length <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> the two diagonals are <a href="/facts/Perpendicular/u0k8xqx4">perpendicular</a>.</li> <li>The two bimedians are perpendicular if and only if the two diagonals have equal length.</li></ul> <h3>Special cases</h3> <p>The Varignon parallelogram is a <a href="/facts/Rhombus/T4pU50GF">rhombus</a> if and only if the two diagonals of the quadrilateral have equal length, that is, if the quadrilateral is an <a href="/facts/Equidiagonal_quadrilateral/hcWuzEtB">equidiagonal quadrilateral</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> </p><p>The Varignon parallelogram is a <a href="/facts/Rectangle/mdLfIGKA">rectangle</a> if and only if the diagonals of the quadrilateral are <a href="/facts/Perpendicular/u0k8xqx4">perpendicular</a>, that is, if the quadrilateral is an <a href="/facts/Orthodiagonal_quadrilateral/R6M3E372">orthodiagonal quadrilateral</a>.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>: p. 14 <a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>: p. 169 </p><p>For a <a href="/facts/List_of_self-intersecting_polygons/zySCnDLM">self-crossing</a> quadrilateral, the Varignon parallelogram can degenerate to four collinear points, forming a line segment traversed twice. This happens whenever the polygon is formed by replacing two parallel sides of a <a href="/facts/Trapezoid/nCBKsauz">trapezoid</a> by the two diagonals of the trapezoid, such as in the <a href="/facts/Antiparallelogram/QDzKogA3">antiparallelogram</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Perpendicular_bisector_construction_of_a_quadrilateral/PeLXTFHI">Perpendicular bisector construction of a quadrilateral</a>, a different way of forming another quadrilateral from a given quadrilateral</li> <li><a href="/facts/Morley%27s_trisector_theorem/2tjYu6oY">Morley's trisector theorem</a>, a related theorem on triangles</li></ul> <h2 id="notes">Notes</h2> <h2 id="references-and-further-reading">References and further reading</h2> <ul><li>H. S. M. Coxeter, S. L. Greitzer: <i>Geometry Revisited</i>. MAA, Washington 1967, pp. 52-54</li> <li>Peter N. Oliver: <a href="https://old.maa.org/sites/default/files/images/upload_library/46/NCTM/mt2001-Varignon2.pdf"><i>Consequences of Varignon Parallelogram Theorem</i></a>. Mathematics Teacher, Band 94, Nr. 5, Mai 2001, pp. 406-408</li></ul> <h2 id="external-links">External links</h2> Wikimedia Commons has media related to Varignon's theorem. <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/VarignonsTheorem.html">"Varignon's theorem"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="http://www.vias.org/comp_geometry/geom_quad_varignon.html">Varignon Parallelogram in Compendium Geometry</a></li> <li><a href="http://dynamicmathematicslearning.com/octagoncentroids.html">A generalization of Varignon's theorem to 2n-gons and to 3D</a> at <a href="http://dynamicmathematicslearning.com/JavaGSPLinks.htm">Dynamic Geometry Sketches</a>, interactive dynamic geometry sketches.</li> <li><a href="http://www.cut-the-knot.org/Curriculum/Geometry/Varignon.shtml">Varignon parallelogram</a> at cut-the-knot-org</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Peter N. Oliver: Pierre Varignon and the Parallelogram Theorem. Mathematics Teacher, Band 94, Nr. 4, April 2001, pp. 316-319 <a href="https://web.archive.org/web/20150906161939/http://www.maa.org/sites/default/files/images/upload_library/46/NCTM/mt2001-Varignon1.pdf" target="_blank">https://web.archive.org/web/20150906161939/http://www.maa.org/sites/default/files/images/upload_library/46/NCTM/mt2001-Varignon1.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Coxeter, H. S. M. and Greitzer, S. L. "Quadrangle; Varignon's theorem" §3.1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 52–54, 1967. <a href="/wiki/Coxeter" target="_blank">/wiki/Coxeter</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Coxeter, H. S. M. and Greitzer, S. L. "Quadrangle; Varignon's theorem" §3.1 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 52–54, 1967. <a href="/wiki/Coxeter" target="_blank">/wiki/Coxeter</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Altshiller-Court, Nathan, College Geometry, Dover Publ., 2007. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"Mateescu Constantin, Answer to Inequality Of Diagonal". Archived from the original on 2014-10-24. Retrieved 2016-04-05. <a href="https://web.archive.org/web/20141024134419/http://www.artofproblemsolving.com/Forum/viewtopic.php?t=363253" target="_blank">https://web.archive.org/web/20141024134419/http://www.artofproblemsolving.com/Forum/viewtopic.php?t=363253</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Altshiller-Court, Nathan, College Geometry, Dover Publ., 2007. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Josefsson, Martin (2011), "The Area of a Bicentric Quadrilateral" (PDF), Forum Geometricorum, 11: 155–164, archived from the original (PDF) on 2020-01-05, retrieved 2016-04-05. <a href="https://web.archive.org/web/20200105031952/http://forumgeom.fau.edu/FG2011volume11/FG201116.pdf" target="_blank">https://web.archive.org/web/20200105031952/http://forumgeom.fau.edu/FG2011volume11/FG201116.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Josefsson, Martin (2012), "Characterizations of Orthodiagonal Quadrilaterals" (PDF), Forum Geometricorum, 12: 13–25, archived from the original (PDF) on 2020-12-05, retrieved 2012-12-28. <a href="https://web.archive.org/web/20201205213638/http://forumgeom.fau.edu/FG2012volume12/FG201202.pdf" target="_blank">https://web.archive.org/web/20201205213638/http://forumgeom.fau.edu/FG2012volume12/FG201202.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>de Villiers, Michael (2009), Some Adventures in Euclidean Geometry, Dynamic Mathematics Learning, p. 58, ISBN 9780557102952. <a href="9780557102952" target="_blank">9780557102952</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Josefsson, Martin (2012), "Characterizations of Orthodiagonal Quadrilaterals" (PDF), Forum Geometricorum, 12: 13–25, archived from the original (PDF) on 2020-12-05, retrieved 2012-12-28. <a href="https://web.archive.org/web/20201205213638/http://forumgeom.fau.edu/FG2012volume12/FG201202.pdf" target="_blank">https://web.archive.org/web/20201205213638/http://forumgeom.fau.edu/FG2012volume12/FG201202.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>de Villiers, Michael (2009), Some Adventures in Euclidean Geometry, Dynamic Mathematics Learning, p. 58, ISBN 9780557102952. <a href="9780557102952" target="_blank">9780557102952</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Muirhead, R. F. (February 1901), "Geometry of the isosceles trapezium and the contra-parallelogram, with applications to the geometry of the ellipse", Proceedings of the Edinburgh Mathematical Society, 20: 70–72, doi:10.1017/s0013091500032892 <a href="/wiki/Robert_Franklin_Muirhead" target="_blank">/wiki/Robert_Franklin_Muirhead</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> </ol>