425:
2489:
2390:
2404:
2397:
2016:
539:
2383:
433:
47:
1740:
2005:
1211:
1438:
780:
1603:
966:
2608:, sometimes called a bounding parallelogram, formed by the tangent lines to the ellipse at the four endpoints of the conjugate diameters. All tangent parallelograms for a given ellipse have the same area.
1598:
1330:
659:
1879:
1550:
1793:
996:
842:
603:
2123:
2075:
1843:
1335:
1273:
873:
2241:
670:
2205:
533:
463:
550:
of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the rectangle is
2281:
Parallelograms can tile the plane by translation. If edges are equal, or angles are right, the symmetry of the lattice is higher. These represent the four
1735:{\displaystyle V={\begin{bmatrix}a_{1}&a_{2}&\dots &a_{n}\\b_{1}&b_{2}&\dots &b_{n}\end{bmatrix}}\in \mathbb {R} ^{2\times n}}
2604:
to the ellipse at an endpoint of one diameter is parallel to the other diameter. Each pair of conjugate diameters of an ellipse has a corresponding
881:
166:
sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The
2612:
2709:
Zalman
Usiskin and Jennifer Griffin, "The Classification of Quadrilaterals. A Study of Definition", Information Age Publishing, 2008, p. 22.
213:– A parallelogram with four sides of equal length. Any parallelogram that is neither a rectangle nor a rhombus was traditionally called a
405:
The centers of four squares all constructed either internally or externally on the sides of a parallelogram are the vertices of a square.
2665:
2894:
155:
309:
The sum of the distances from any interior point to the sides is independent of the location of the point. (This is an extension of
1558:
1290:
3424:
174:
and neither condition can be proven without appealing to the
Euclidean parallel postulate or one of its equivalent formulations.
2720:
2000:{\displaystyle K=\left|{\begin{matrix}a_{1}&a_{2}&1\\b_{1}&b_{2}&1\\c_{1}&c_{2}&1\end{matrix}}\right|.}
614:
2353:
89:
1459:
2645:
1206:{\displaystyle K=2{\sqrt {S(S-B)(S-C)(S-D_{1})}}={\frac {1}{2}}{\sqrt {(B+C+D_{1})(-B+C+D_{1})(B-C+D_{1})(B+C-D_{1})}},}
376:
then it must be a rhombus or an oblong (a non-square rectangle). If it has four lines of reflectional symmetry, it is a
1753:
3017:
2997:
2813:
2515:(that is, not self-intersecting), then the area of the Varignon parallelogram is half the area of the quadrilateral.
799:
556:
3429:
2992:
2949:
2924:
402:
Unlike any other convex polygon, a parallelogram cannot be inscribed in any triangle with less than twice its area.
2837:
2084:
2036:
3052:
1801:
2977:
2842:
1433:{\displaystyle V={\begin{bmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{bmatrix}}\in \mathbb {R} ^{2\times 2}}
352:
2738:
3002:
2887:
31:
3403:
3343:
2982:
2605:
2024:
542:
The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram
373:
362:
282:
167:
2496:
2483:
2015:
2851:
3287:
3057:
2987:
2929:
1275:
and the leading factor 2 comes from the fact that the chosen diagonal divides the parallelogram into
1219:
348:
310:
275:
3393:
3368:
3338:
3333:
3292:
3007:
2597:
2518:
2423:
2133:
412:
to a diagonal, then the parallelograms formed on opposite sides of that diagonal are equal in area.
369:
303:
265:
163:
3398:
2939:
332:, if just one of these statements is true in a simple quadrilateral, then it is a parallelogram.
171:
143:
987:
775:{\displaystyle K=K_{\text{rect}}-2\times K_{\text{tri}}=((B+A)\times H)-(A\times H)=B\times H.}
3378:
2972:
2880:
2820:
2640:
2427:
432:
296:
220:
79:
2676:
858:
2907:
2693:
2211:
409:
344:
The area of a parallelogram is twice the area of a triangle created by one of its diagonals.
329:
94:
2764:
2503:
of the sides of an arbitrary quadrilateral are the vertices of a parallelogram, called its
2178:
546:
The base × height area formula can also be derived using the figure to the right. The area
506:
341:
Opposite sides of a parallelogram are parallel (by definition) and so will never intersect.
3373:
3353:
3348:
3318:
3037:
3012:
2944:
2628:
2555:(each of the two pairs reconstructs the quadrilateral) while that of the small triangles,
2512:
2282:
439:
271:
223:– A parallelogram with four sides of equal length and angles of equal size (right angles).
75:
847:
Provided that the parallelogram is not a rhombus, the area can be expressed using sides
372:
of order 2 (through 180°) (or order 4 if a square). If it also has exactly two lines of
3383:
3363:
3328:
3323:
2954:
2934:
2624:
2508:
2488:
489:
241:
233:
185:
151:
134:
2389:
3418:
3358:
3209:
3102:
2964:
500:
of a parallelogram is the same as that of a rectangle with the same base and height:
470:
320:
in the plane of the quadrilateral with the property that every straight line through
292:
237:
159:
64:
2866:
2615:
an ellipse from any pair of conjugate diameters, or from any tangent parallelogram.
2403:
2396:
3388:
3258:
3214:
3178:
3168:
3163:
2855:
2846:
2823:
2601:
2569:(half linear dimensions yields quarter area), and the area of the parallelogram is
2785:
2382:
2254:
divide each other into segments of equal length, the diagonals bisect each other.
538:
3297:
3204:
3183:
3173:
2860:
1846:
424:
2814:
Parallelogram and
Rhombus - Animated course (Construction, Circumference, Area)
170:
of opposite sides and opposite angles is a direct consequence of the
Euclidean
3302:
3158:
3148:
3032:
2023:
To prove that the diagonals of a parallelogram bisect each other, we will use
17:
2534:
The base pairs form a parallelogram with half the area of the quadrilateral,
415:
The diagonals of a parallelogram divide it into four triangles of equal area.
177:
By comparison, a quadrilateral with at least one pair of parallel sides is a
30:
This article is about the quadrilateral. For the album by Linda
Perhacs, see
3277:
3267:
3244:
3234:
3224:
3153:
3062:
3027:
2828:
2784:
Weisstein, Eric W. "Parallelogram." From MathWorld--A Wolfram Web
Resource.
961:{\displaystyle K={\frac {|\tan \gamma |}{2}}\cdot \left|B^{2}-C^{2}\right|.}
493:
485:
204:
178:
68:
2430:
are in the same proportions as its sides (though in a different order). If
126:(product of adjacent sides and sine of the vertex angle determined by them)
46:
473:
apply to parallelograms. Further formulas are specific to parallelograms:
295:
of the sides equals the sum of the squares of the diagonals. (This is the
3282:
3272:
3229:
3188:
3117:
3107:
3097:
2916:
2500:
2447:
2027:
285:
258:
214:
52:
3239:
3219:
3132:
3127:
3122:
3112:
3087:
3042:
2903:
2593:
210:
428:
A parallelogram can be rearranged into a rectangle with the same area.
3047:
377:
2718:
Chen, Zhibo, and Liang, Tian. "The converse of
Viviani's theorem",
3092:
2872:
2487:
537:
408:
If two lines parallel to sides of a parallelogram are constructed
358:
Any line through the midpoint of a parallelogram bisects the area.
347:
The area of a parallelogram is also equal to the magnitude of the
328:
Thus all parallelograms have all the properties listed above, and
2666:"CIMT - Page no longer available at Plymouth University servers"
497:
207:– A parallelogram with four angles of equal size (right angles).
105:
2876:
2154:, since opposite sides of a parallelogram are equal in length.
2528:
Bases of similar triangles are parallel to the blue diagonal.
1593:{\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{n}}
1325:{\displaystyle \mathbf {a} ,\mathbf {b} \in \mathbb {R} ^{2}}
2014:
1873:
as rows with the last column padded using ones as follows:
496:, as shown in the figure to the left. This means that the
184:
The three-dimensional counterpart of a parallelogram is a
2867:
Interactive applet showing parallelogram area calculation
1861:
is equivalent to the determinant of a matrix built using
324:
divides the quadrilateral into two regions of equal area.
248:
Two pairs of opposite sides are parallel (by definition).
654:{\displaystyle K_{\text{tri}}={\frac {A}{2}}\times H.\,}
2700:, Mathematical Association of America, 2010, pp. 51-52.
2541:, as the sum of the areas of the four large triangles,
181:
in
American English or a trapezium in British English.
2132:(since these are angles that a transversal makes with
1894:
1618:
1350:
2796:
Mitchell, Douglas W., "The area of a quadrilateral",
2214:
2181:
2087:
2039:
1882:
1804:
1756:
1606:
1561:
1462:
1338:
1293:
1222:
999:
971:
When the parallelogram is specified from the lengths
884:
861:
802:
673:
617:
559:
509:
442:
986:
of either diagonal, then the area can be found from
3311:
3257:
3197:
3141:
3080:
3071:
2963:
2915:
2838:
2749:Dunn, J.A., and J.E. Pretty, "Halving a triangle",
1545:{\displaystyle |\det(V)|=|a_{1}b_{2}-a_{2}b_{1}|\,}
130:
104:
88:
74:
60:
39:
2235:
2199:
2117:
2069:
1999:
1837:
1787:
1742:. Then the area of the parallelogram generated by
1734:
1592:
1544:
1448:. Then the area of the parallelogram generated by
1432:
1324:
1267:
1205:
960:
867:
836:
774:
653:
597:
527:
457:
254:Two pairs of opposite angles are equal in measure.
2852:Equilateral Triangles On Sides of a Parallelogram
1788:{\displaystyle {\sqrt {\det(VV^{\mathrm {T} })}}}
281:Each diagonal divides the quadrilateral into two
191:The word comes from the Greek παραλληλό-γραμμον,
2126:(alternate interior angles are equal in measure)
2078:(alternate interior angles are equal in measure)
1759:
1468:
251:Two pairs of opposite sides are equal in length.
217:but this term is not used in modern mathematics.
2786:http://mathworld.wolfram.com/Parallelogram.html
979:of two adjacent sides together with the length
837:{\displaystyle K=B\cdot C\cdot \sin \theta .\,}
598:{\displaystyle K_{\text{rect}}=(B+A)\times H\,}
471:area formulas for general convex quadrilaterals
365:takes a parallelogram to another parallelogram.
2167:two corresponding angles and the included side
27:Quadrilateral with two pairs of parallel sides
2888:
2525:An arbitrary quadrilateral and its diagonals.
244:any one of the following statements is true:
8:
2861:Definition and properties of a parallelogram
664:Therefore, the area of the parallelogram is
55:as it has unequal sides and no right angles.
2780:
2778:
195:, which means a shape "of parallel lines".
3077:
2895:
2881:
2873:
2492:Proof without words of Varignon's theorem
2434:is an automedian triangle in which vertex
2118:{\displaystyle \angle BAE\cong \angle DCE}
2070:{\displaystyle \angle ABE\cong \angle CDE}
2413:Parallelograms arising from other figures
2213:
2180:
2086:
2038:
1975:
1963:
1944:
1932:
1913:
1901:
1893:
1881:
1838:{\displaystyle a,b,c\in \mathbb {R} ^{2}}
1829:
1825:
1824:
1803:
1773:
1772:
1757:
1755:
1720:
1716:
1715:
1697:
1680:
1668:
1654:
1637:
1625:
1613:
1605:
1584:
1580:
1579:
1570:
1562:
1560:
1541:
1536:
1530:
1520:
1507:
1497:
1488:
1480:
1463:
1461:
1418:
1414:
1413:
1395:
1383:
1369:
1357:
1345:
1337:
1316:
1312:
1311:
1302:
1294:
1292:
1257:
1248:
1221:
1189:
1161:
1133:
1102:
1081:
1071:
1057:
1009:
998:
944:
931:
908:
894:
891:
883:
860:
833:
801:
703:
684:
672:
650:
631:
622:
616:
594:
564:
558:
508:
441:
2627:is a three-dimensional figure whose six
2287:
431:
423:
2657:
2011:Proof that diagonals bisect each other
1849:of the parallelogram with vertices at
875:at the intersection of the diagonals:
383:The perimeter of a parallelogram is 2(
36:
608:and the area of a single triangle is
7:
785:Another area formula, for two sides
2588:Tangent parallelogram of an ellipse
1440:denote the matrix with elements of
2735:2006 British Mathematical Olympiad
2458:is one of the extended medians of
2273:is the midpoint of each diagonal.
2103:
2088:
2055:
2040:
1774:
399:are the lengths of adjacent sides.
25:
2402:
2395:
2388:
2381:
2283:Bravais lattices in 2 dimensions
2257:Separately, since the diagonals
1571:
1563:
1303:
1295:
45:
2721:The College Mathematics Journal
2596:, two diameters are said to be
1268:{\displaystyle S=(B+C+D_{1})/2}
436:Animation for the area formula
2698:Methods for Euclidean Geometry
2692:Owen Byer, Felix Lazebnik and
2165:are congruent (ASA postulate,
1780:
1762:
1537:
1489:
1481:
1477:
1471:
1464:
1254:
1229:
1195:
1170:
1167:
1142:
1139:
1111:
1108:
1083:
1063:
1044:
1041:
1029:
1026:
1014:
909:
895:
754:
742:
736:
727:
715:
712:
585:
573:
264:One pair of opposite sides is
1:
2466:lying on the circumcircle of
2646:Levi-Civita parallelogramoid
2450:(where the three medians of
2376:
2351:
2334:
2531:Ditto for the red diagonal.
2265:bisect each other at point
2150:is equal in length to side
3446:
2507:. If the quadrilateral is
2481:
476:A parallelogram with base
29:
2765:"Triangle Circumscribing"
2724:37(5), 2006, pp. 390–391.
2619:Faces of a parallelepiped
2438:stands opposite the side
2363:
2277:Lattice of parallelograms
44:
492:, and rearranged into a
236:(non-self-intersecting)
51:This parallelogram is a
3425:Types of quadrilaterals
2673:www.cimt.plymouth.ac.uk
1283:From vertex coordinates
868:{\displaystyle \gamma }
2505:Varignon parallelogram
2493:
2478:Varignon parallelogram
2237:
2236:{\displaystyle BE=DE.}
2201:
2119:
2071:
2020:
2001:
1839:
1789:
1736:
1594:
1546:
1434:
1326:
1269:
1207:
962:
869:
838:
776:
655:
599:
543:
529:
484:can be divided into a
466:
459:
429:
32:Parallelograms (album)
2843:Area of Parallelogram
2753:56, May 1972, p. 105.
2606:tangent parallelogram
2491:
2238:
2202:
2200:{\displaystyle AE=CE}
2157:Therefore, triangles
2120:
2072:
2018:
2002:
1840:
1790:
1737:
1595:
1547:
1435:
1327:
1279:congruent triangles.
1270:
1208:
990:. Specifically it is
963:
870:
839:
777:
656:
600:
541:
530:
528:{\displaystyle K=bh.}
460:
435:
427:
374:reflectional symmetry
363:affine transformation
3128:Nonagon/Enneagon (9)
3058:Tangential trapezoid
2863:with animated applet
2798:Mathematical Gazette
2751:Mathematical Gazette
2631:are parallelograms.
2474:is a parallelogram.
2324:Centered rectangular
2246:Since the diagonals
2212:
2179:
2085:
2037:
1880:
1802:
1754:
1604:
1559:
1460:
1336:
1291:
1220:
997:
882:
859:
800:
671:
615:
557:
507:
458:{\displaystyle K=bh}
440:
368:A parallelogram has
349:vector cross product
268:and equal in length.
3240:Megagon (1,000,000)
3008:Isosceles trapezoid
2763:Weisstein, Eric W.
2600:if and only if the
2519:Proof without words
2424:automedian triangle
2418:Automedian triangle
2290:
370:rotational symmetry
361:Any non-degenerate
304:rotational symmetry
240:is a parallelogram
3210:Icositetragon (24)
2869:interactive applet
2821:Weisstein, Eric W.
2769:Wolfram Math World
2611:It is possible to
2497:Varignon's theorem
2494:
2484:Varignon's theorem
2288:
2233:
2197:
2115:
2067:
2021:
2019:Parallelogram ABCD
1997:
1988:
1835:
1785:
1732:
1705:
1590:
1542:
1430:
1403:
1322:
1265:
1203:
958:
865:
834:
772:
651:
595:
544:
525:
467:
455:
430:
261:bisect each other.
172:parallel postulate
162:with two pairs of
144:Euclidean geometry
3430:Elementary shapes
3412:
3411:
3253:
3252:
3230:Myriagon (10,000)
3215:Triacontagon (30)
3179:Heptadecagon (17)
3169:Pentadecagon (15)
3164:Tetradecagon (14)
3103:Quadrilateral (4)
2973:Antiparallelogram
2641:Antiparallelogram
2410:
2409:
1783:
1198:
1079:
1066:
917:
706:
687:
639:
625:
567:
316:There is a point
311:Viviani's theorem
297:parallelogram law
228:Characterizations
193:parallēló-grammon
156:self-intersecting
140:
139:
16:(Redirected from
3437:
3225:Chiliagon (1000)
3205:Icositrigon (23)
3184:Octadecagon (18)
3174:Hexadecagon (16)
3078:
2897:
2890:
2883:
2874:
2834:
2833:
2801:
2794:
2788:
2782:
2773:
2772:
2760:
2754:
2747:
2741:
2731:
2725:
2716:
2710:
2707:
2701:
2694:Deirdre Smeltzer
2690:
2684:
2683:
2681:
2675:. Archived from
2670:
2662:
2562:is a quarter of
2454:intersect), and
2406:
2399:
2392:
2385:
2339:α=90°, a=b
2291:
2242:
2240:
2239:
2234:
2206:
2204:
2203:
2198:
2124:
2122:
2121:
2116:
2076:
2074:
2073:
2068:
2006:
2004:
2003:
1998:
1993:
1989:
1980:
1979:
1968:
1967:
1949:
1948:
1937:
1936:
1918:
1917:
1906:
1905:
1844:
1842:
1841:
1836:
1834:
1833:
1828:
1794:
1792:
1791:
1786:
1784:
1779:
1778:
1777:
1758:
1741:
1739:
1738:
1733:
1731:
1730:
1719:
1710:
1709:
1702:
1701:
1685:
1684:
1673:
1672:
1659:
1658:
1642:
1641:
1630:
1629:
1599:
1597:
1596:
1591:
1589:
1588:
1583:
1574:
1566:
1551:
1549:
1548:
1543:
1540:
1535:
1534:
1525:
1524:
1512:
1511:
1502:
1501:
1492:
1484:
1467:
1439:
1437:
1436:
1431:
1429:
1428:
1417:
1408:
1407:
1400:
1399:
1388:
1387:
1374:
1373:
1362:
1361:
1331:
1329:
1328:
1323:
1321:
1320:
1315:
1306:
1298:
1274:
1272:
1271:
1266:
1261:
1253:
1252:
1212:
1210:
1209:
1204:
1199:
1194:
1193:
1166:
1165:
1138:
1137:
1107:
1106:
1082:
1080:
1072:
1067:
1062:
1061:
1010:
967:
965:
964:
959:
954:
950:
949:
948:
936:
935:
918:
913:
912:
898:
892:
874:
872:
871:
866:
843:
841:
840:
835:
793:and angle θ, is
781:
779:
778:
773:
708:
707:
704:
689:
688:
685:
660:
658:
657:
652:
640:
632:
627:
626:
623:
604:
602:
601:
596:
569:
568:
565:
534:
532:
531:
526:
464:
462:
461:
456:
336:Other properties
117:(base × height);
49:
37:
21:
3445:
3444:
3440:
3439:
3438:
3436:
3435:
3434:
3415:
3414:
3413:
3408:
3307:
3261:
3249:
3193:
3159:Tridecagon (13)
3149:Hendecagon (11)
3137:
3073:
3067:
3038:Right trapezoid
2959:
2911:
2901:
2824:"Parallelogram"
2819:
2818:
2810:
2805:
2804:
2795:
2791:
2783:
2776:
2762:
2761:
2757:
2748:
2744:
2732:
2728:
2717:
2713:
2708:
2704:
2691:
2687:
2679:
2668:
2664:
2663:
2659:
2654:
2637:
2621:
2590:
2581:
2574:
2567:
2560:
2553:
2546:
2539:
2499:holds that the
2486:
2480:
2420:
2415:
2330:
2326:(orthorhombic)
2325:
2321:(orthorhombic)
2320:
2315:
2279:
2210:
2209:
2177:
2176:
2083:
2082:
2035:
2034:
2013:
1987:
1986:
1981:
1971:
1969:
1959:
1956:
1955:
1950:
1940:
1938:
1928:
1925:
1924:
1919:
1909:
1907:
1897:
1889:
1878:
1877:
1823:
1800:
1799:
1768:
1752:
1751:
1714:
1704:
1703:
1693:
1691:
1686:
1676:
1674:
1664:
1661:
1660:
1650:
1648:
1643:
1633:
1631:
1621:
1614:
1602:
1601:
1578:
1557:
1556:
1526:
1516:
1503:
1493:
1458:
1457:
1412:
1402:
1401:
1391:
1389:
1379:
1376:
1375:
1365:
1363:
1353:
1346:
1334:
1333:
1310:
1289:
1288:
1285:
1244:
1218:
1217:
1185:
1157:
1129:
1098:
1053:
995:
994:
988:Heron's formula
985:
940:
927:
926:
922:
893:
880:
879:
857:
856:
798:
797:
699:
680:
669:
668:
618:
613:
612:
560:
555:
554:
505:
504:
438:
437:
422:
338:
291:The sum of the
272:Adjacent angles
230:
201:
118:
98:
56:
35:
28:
23:
22:
15:
12:
11:
5:
3443:
3441:
3433:
3432:
3427:
3417:
3416:
3410:
3409:
3407:
3406:
3401:
3396:
3391:
3386:
3381:
3376:
3371:
3366:
3364:Pseudotriangle
3361:
3356:
3351:
3346:
3341:
3336:
3331:
3326:
3321:
3315:
3313:
3309:
3308:
3306:
3305:
3300:
3295:
3290:
3285:
3280:
3275:
3270:
3264:
3262:
3255:
3254:
3251:
3250:
3248:
3247:
3242:
3237:
3232:
3227:
3222:
3217:
3212:
3207:
3201:
3199:
3195:
3194:
3192:
3191:
3186:
3181:
3176:
3171:
3166:
3161:
3156:
3154:Dodecagon (12)
3151:
3145:
3143:
3139:
3138:
3136:
3135:
3130:
3125:
3120:
3115:
3110:
3105:
3100:
3095:
3090:
3084:
3082:
3075:
3069:
3068:
3066:
3065:
3060:
3055:
3050:
3045:
3040:
3035:
3030:
3025:
3020:
3015:
3010:
3005:
3000:
2995:
2990:
2985:
2980:
2975:
2969:
2967:
2965:Quadrilaterals
2961:
2960:
2958:
2957:
2952:
2947:
2942:
2937:
2932:
2927:
2921:
2919:
2913:
2912:
2902:
2900:
2899:
2892:
2885:
2877:
2871:
2870:
2864:
2858:
2849:
2840:
2835:
2816:
2809:
2808:External links
2806:
2803:
2802:
2789:
2774:
2755:
2742:
2726:
2711:
2702:
2685:
2682:on 2014-05-14.
2656:
2655:
2653:
2650:
2649:
2648:
2643:
2636:
2633:
2625:parallelepiped
2620:
2617:
2589:
2586:
2585:
2584:
2579:
2572:
2565:
2558:
2551:
2544:
2537:
2532:
2529:
2526:
2521:(see figure):
2482:Main article:
2479:
2476:
2419:
2416:
2414:
2411:
2408:
2407:
2400:
2393:
2386:
2379:
2375:
2374:
2368:
2364:pmm, , order 4
2362:
2358:p4m, , order 8
2356:
2350:
2349:
2346:
2343:
2340:
2337:
2333:
2332:
2327:
2322:
2317:
2312:
2308:
2307:
2304:
2301:
2298:
2295:
2278:
2275:
2244:
2243:
2232:
2229:
2226:
2223:
2220:
2217:
2207:
2196:
2193:
2190:
2187:
2184:
2134:parallel lines
2130:
2129:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2080:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2012:
2009:
2008:
2007:
1996:
1992:
1985:
1982:
1978:
1974:
1970:
1966:
1962:
1958:
1957:
1954:
1951:
1947:
1943:
1939:
1935:
1931:
1927:
1926:
1923:
1920:
1916:
1912:
1908:
1904:
1900:
1896:
1895:
1892:
1888:
1885:
1832:
1827:
1822:
1819:
1816:
1813:
1810:
1807:
1782:
1776:
1771:
1767:
1764:
1761:
1729:
1726:
1723:
1718:
1713:
1708:
1700:
1696:
1692:
1690:
1687:
1683:
1679:
1675:
1671:
1667:
1663:
1662:
1657:
1653:
1649:
1647:
1644:
1640:
1636:
1632:
1628:
1624:
1620:
1619:
1617:
1612:
1609:
1587:
1582:
1577:
1573:
1569:
1565:
1539:
1533:
1529:
1523:
1519:
1515:
1510:
1506:
1500:
1496:
1491:
1487:
1483:
1479:
1476:
1473:
1470:
1466:
1427:
1424:
1421:
1416:
1411:
1406:
1398:
1394:
1390:
1386:
1382:
1378:
1377:
1372:
1368:
1364:
1360:
1356:
1352:
1351:
1349:
1344:
1341:
1319:
1314:
1309:
1305:
1301:
1297:
1284:
1281:
1264:
1260:
1256:
1251:
1247:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1214:
1213:
1202:
1197:
1192:
1188:
1184:
1181:
1178:
1175:
1172:
1169:
1164:
1160:
1156:
1153:
1150:
1147:
1144:
1141:
1136:
1132:
1128:
1125:
1122:
1119:
1116:
1113:
1110:
1105:
1101:
1097:
1094:
1091:
1088:
1085:
1078:
1075:
1070:
1065:
1060:
1056:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1016:
1013:
1008:
1005:
1002:
983:
969:
968:
957:
953:
947:
943:
939:
934:
930:
925:
921:
916:
911:
907:
904:
901:
897:
890:
887:
864:
845:
844:
832:
829:
826:
823:
820:
817:
814:
811:
808:
805:
783:
782:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
741:
738:
735:
732:
729:
726:
723:
720:
717:
714:
711:
702:
698:
695:
692:
683:
679:
676:
662:
661:
649:
646:
643:
638:
635:
630:
621:
606:
605:
593:
590:
587:
584:
581:
578:
575:
572:
563:
536:
535:
524:
521:
518:
515:
512:
490:right triangle
454:
451:
448:
445:
421:
418:
417:
416:
413:
406:
403:
400:
381:
366:
359:
356:
345:
342:
337:
334:
326:
325:
314:
307:
300:
289:
279:
269:
262:
255:
252:
249:
242:if and only if
229:
226:
225:
224:
218:
208:
200:
197:
186:parallelepiped
138:
137:
132:
128:
127:
108:
102:
101:
96:
92:
90:Symmetry group
86:
85:
82:
72:
71:
62:
58:
57:
50:
42:
41:
26:
24:
18:Parallelograms
14:
13:
10:
9:
6:
4:
3:
2:
3442:
3431:
3428:
3426:
3423:
3422:
3420:
3405:
3404:Weakly simple
3402:
3400:
3397:
3395:
3392:
3390:
3387:
3385:
3382:
3380:
3377:
3375:
3372:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3352:
3350:
3347:
3345:
3344:Infinite skew
3342:
3340:
3337:
3335:
3332:
3330:
3327:
3325:
3322:
3320:
3317:
3316:
3314:
3310:
3304:
3301:
3299:
3296:
3294:
3291:
3289:
3286:
3284:
3281:
3279:
3276:
3274:
3271:
3269:
3266:
3265:
3263:
3260:
3259:Star polygons
3256:
3246:
3245:Apeirogon (∞)
3243:
3241:
3238:
3236:
3233:
3231:
3228:
3226:
3223:
3221:
3218:
3216:
3213:
3211:
3208:
3206:
3203:
3202:
3200:
3196:
3190:
3189:Icosagon (20)
3187:
3185:
3182:
3180:
3177:
3175:
3172:
3170:
3167:
3165:
3162:
3160:
3157:
3155:
3152:
3150:
3147:
3146:
3144:
3140:
3134:
3131:
3129:
3126:
3124:
3121:
3119:
3116:
3114:
3111:
3109:
3106:
3104:
3101:
3099:
3096:
3094:
3091:
3089:
3086:
3085:
3083:
3079:
3076:
3070:
3064:
3061:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3029:
3026:
3024:
3023:Parallelogram
3021:
3019:
3018:Orthodiagonal
3016:
3014:
3011:
3009:
3006:
3004:
3001:
2999:
2998:Ex-tangential
2996:
2994:
2991:
2989:
2986:
2984:
2981:
2979:
2976:
2974:
2971:
2970:
2968:
2966:
2962:
2956:
2953:
2951:
2948:
2946:
2943:
2941:
2938:
2936:
2933:
2931:
2928:
2926:
2923:
2922:
2920:
2918:
2914:
2909:
2905:
2898:
2893:
2891:
2886:
2884:
2879:
2878:
2875:
2868:
2865:
2862:
2859:
2857:
2853:
2850:
2848:
2844:
2841:
2839:
2836:
2831:
2830:
2825:
2822:
2817:
2815:
2812:
2811:
2807:
2799:
2793:
2790:
2787:
2781:
2779:
2775:
2770:
2766:
2759:
2756:
2752:
2746:
2743:
2739:
2736:
2730:
2727:
2723:
2722:
2715:
2712:
2706:
2703:
2699:
2695:
2689:
2686:
2678:
2674:
2667:
2661:
2658:
2651:
2647:
2644:
2642:
2639:
2638:
2634:
2632:
2630:
2626:
2618:
2616:
2614:
2609:
2607:
2603:
2599:
2595:
2587:
2582:
2575:
2568:
2561:
2554:
2547:
2540:
2533:
2530:
2527:
2524:
2523:
2522:
2520:
2516:
2514:
2510:
2506:
2502:
2498:
2490:
2485:
2477:
2475:
2473:
2469:
2465:
2461:
2457:
2453:
2449:
2445:
2441:
2437:
2433:
2429:
2426:is one whose
2425:
2417:
2412:
2405:
2401:
2398:
2394:
2391:
2387:
2384:
2380:
2377:
2373:
2370:p1, , order 2
2369:
2367:
2361:
2357:
2355:
2352:
2347:
2344:
2341:
2338:
2335:
2331:(monoclinic)
2328:
2323:
2318:
2316:(tetragonal)
2313:
2310:
2309:
2305:
2302:
2299:
2296:
2293:
2292:
2286:
2284:
2276:
2274:
2272:
2268:
2264:
2260:
2255:
2253:
2249:
2230:
2227:
2224:
2221:
2218:
2215:
2208:
2194:
2191:
2188:
2185:
2182:
2175:
2174:
2173:
2170:
2168:
2164:
2160:
2155:
2153:
2149:
2144:
2142:
2138:
2135:
2127:
2112:
2109:
2106:
2100:
2097:
2094:
2091:
2081:
2079:
2064:
2061:
2058:
2052:
2049:
2046:
2043:
2033:
2032:
2031:
2029:
2026:
2017:
2010:
1994:
1990:
1983:
1976:
1972:
1964:
1960:
1952:
1945:
1941:
1933:
1929:
1921:
1914:
1910:
1902:
1898:
1890:
1886:
1883:
1876:
1875:
1874:
1872:
1868:
1864:
1860:
1856:
1852:
1848:
1830:
1820:
1817:
1814:
1811:
1808:
1805:
1796:
1769:
1765:
1749:
1745:
1727:
1724:
1721:
1711:
1706:
1698:
1694:
1688:
1681:
1677:
1669:
1665:
1655:
1651:
1645:
1638:
1634:
1626:
1622:
1615:
1610:
1607:
1585:
1575:
1567:
1553:
1531:
1527:
1521:
1517:
1513:
1508:
1504:
1498:
1494:
1485:
1474:
1455:
1451:
1447:
1443:
1425:
1422:
1419:
1409:
1404:
1396:
1392:
1384:
1380:
1370:
1366:
1358:
1354:
1347:
1342:
1339:
1317:
1307:
1299:
1282:
1280:
1278:
1262:
1258:
1249:
1245:
1241:
1238:
1235:
1232:
1226:
1223:
1200:
1190:
1186:
1182:
1179:
1176:
1173:
1162:
1158:
1154:
1151:
1148:
1145:
1134:
1130:
1126:
1123:
1120:
1117:
1114:
1103:
1099:
1095:
1092:
1089:
1086:
1076:
1073:
1068:
1058:
1054:
1050:
1047:
1038:
1035:
1032:
1023:
1020:
1017:
1011:
1006:
1003:
1000:
993:
992:
991:
989:
982:
978:
974:
955:
951:
945:
941:
937:
932:
928:
923:
919:
914:
905:
902:
899:
888:
885:
878:
877:
876:
862:
854:
850:
830:
827:
824:
821:
818:
815:
812:
809:
806:
803:
796:
795:
794:
792:
788:
769:
766:
763:
760:
757:
751:
748:
745:
739:
733:
730:
724:
721:
718:
709:
700:
696:
693:
690:
681:
677:
674:
667:
666:
665:
647:
644:
641:
636:
633:
628:
619:
611:
610:
609:
591:
588:
582:
579:
576:
570:
561:
553:
552:
551:
549:
540:
522:
519:
516:
513:
510:
503:
502:
501:
499:
495:
491:
487:
483:
479:
474:
472:
452:
449:
446:
443:
434:
426:
419:
414:
411:
407:
404:
401:
398:
394:
390:
386:
382:
379:
375:
371:
367:
364:
360:
357:
354:
350:
346:
343:
340:
339:
335:
333:
331:
323:
319:
315:
312:
308:
305:
301:
298:
294:
290:
287:
284:
280:
277:
276:supplementary
273:
270:
267:
263:
260:
256:
253:
250:
247:
246:
245:
243:
239:
238:quadrilateral
235:
227:
222:
219:
216:
212:
209:
206:
203:
202:
199:Special cases
198:
196:
194:
189:
187:
182:
180:
175:
173:
169:
165:
161:
160:quadrilateral
157:
153:
149:
148:parallelogram
145:
136:
133:
129:
125:
121:
116:
112:
109:
107:
103:
99:
93:
91:
87:
83:
81:
77:
73:
70:
66:
65:quadrilateral
63:
59:
54:
48:
43:
40:Parallelogram
38:
33:
19:
3198:>20 sides
3133:Decagon (10)
3118:Heptagon (7)
3108:Pentagon (5)
3098:Triangle (3)
3022:
2993:Equidiagonal
2856:cut-the-knot
2847:cut-the-knot
2827:
2800:, July 2009.
2797:
2792:
2768:
2758:
2750:
2745:
2734:
2729:
2719:
2714:
2705:
2697:
2688:
2677:the original
2672:
2660:
2622:
2610:
2602:tangent line
2591:
2577:
2570:
2563:
2556:
2549:
2542:
2535:
2517:
2504:
2495:
2471:
2467:
2463:
2459:
2455:
2451:
2443:
2439:
2435:
2431:
2421:
2371:
2365:
2359:
2336:Constraints
2280:
2270:
2266:
2262:
2258:
2256:
2251:
2247:
2245:
2171:
2166:
2162:
2158:
2156:
2151:
2147:
2145:
2140:
2136:
2131:
2125:
2077:
2022:
1870:
1866:
1862:
1858:
1854:
1850:
1797:
1750:is equal to
1747:
1743:
1555:Let vectors
1554:
1456:is equal to
1453:
1449:
1445:
1441:
1287:Let vectors
1286:
1276:
1215:
980:
976:
972:
970:
852:
848:
846:
790:
786:
784:
663:
607:
547:
545:
481:
477:
475:
468:
420:Area formula
396:
392:
388:
384:
327:
321:
317:
231:
192:
190:
183:
176:
147:
141:
123:
119:
114:
110:
3394:Star-shaped
3369:Rectilinear
3339:Equilateral
3334:Equiangular
3298:Hendecagram
3142:11–20 sides
3123:Octagon (8)
3113:Hexagon (6)
3088:Monogon (1)
2930:Equilateral
2733:Problem 5,
2613:reconstruct
2342:α=90°
2319:Rectangular
2172:Therefore,
2146:Also, side
1847:signed area
1845:. Then the
1798:Let points
480:and height
469:All of the
306:of order 2.
3419:Categories
3399:Tangential
3303:Dodecagram
3081:1–10 sides
3072:By number
3053:Tangential
3033:Right kite
2652:References
855:and angle
410:concurrent
330:conversely
168:congruence
131:Properties
3379:Reinhardt
3288:Enneagram
3278:Heptagram
3268:Pentagram
3235:65537-gon
3093:Digon (2)
3063:Trapezoid
3028:Rectangle
2978:Bicentric
2940:Isosceles
2917:Triangles
2829:MathWorld
2598:conjugate
2501:midpoints
2306:Rhomboid
2300:Rectangle
2289:Lattices
2104:∠
2101:≅
2089:∠
2056:∠
2053:≅
2041:∠
2028:triangles
2025:congruent
1821:∈
1725:×
1712:∈
1689:…
1646:…
1576:∈
1514:−
1423:×
1410:∈
1308:∈
1183:−
1149:−
1115:−
1051:−
1036:−
1021:−
938:−
920:⋅
906:γ
903:
863:γ
828:θ
825:
819:⋅
813:⋅
764:×
749:×
740:−
731:×
697:×
691:−
642:×
589:×
494:rectangle
486:trapezoid
286:triangles
283:congruent
259:diagonals
205:Rectangle
179:trapezoid
69:trapezium
3354:Isotoxal
3349:Isogonal
3293:Decagram
3283:Octagram
3273:Hexagram
3074:of sides
3003:Harmonic
2904:Polygons
2635:See also
2448:centroid
2354:Symmetry
2269:, point
1600:and let
1332:and let
391:) where
353:adjacent
266:parallel
215:rhomboid
164:parallel
80:vertices
53:rhomboid
3374:Regular
3319:Concave
3312:Classes
3220:257-gon
3043:Rhombus
2983:Crossed
2594:ellipse
2592:For an
2513:concave
2470:, then
2446:is the
2428:medians
2329:Oblique
2311:System
2303:Rhombus
351:of two
302:It has
293:squares
211:Rhombus
3384:Simple
3329:Cyclic
3324:Convex
3048:Square
2988:Cyclic
2950:Obtuse
2945:Kepler
2576:minus
2509:convex
2314:Square
2297:Square
1216:where
488:and a
378:square
355:sides.
234:simple
221:Square
152:simple
135:convex
3359:Magic
2955:Right
2935:Ideal
2925:Acute
2680:(PDF)
2669:(PDF)
2629:faces
2548:is 2
2462:with
2378:Form
2348:None
154:(non-
150:is a
76:Edges
3389:Skew
3013:Kite
2908:List
2472:BGCL
2345:a=b
2294:Form
2261:and
2250:and
2161:and
2139:and
1869:and
1857:and
1746:and
1452:and
1444:and
975:and
851:and
789:and
686:rect
566:rect
498:area
395:and
274:are
257:The
146:, a
122:sin
106:Area
78:and
61:Type
2854:at
2845:at
2511:or
2468:ABC
2460:ABC
2452:ABC
2442:,
2432:ABC
2422:An
2169:).
2163:CDE
2159:ABE
2143:).
1760:det
1469:det
1277:two
900:tan
822:sin
705:tri
624:tri
142:In
100:, ,
3421::
2826:.
2777:^
2767:.
2737:,
2696:,
2671:.
2623:A
2456:AL
2285:.
2263:BD
2259:AC
2252:BD
2248:AC
2152:DC
2148:AB
2141:DC
2137:AB
2030::
1865:,
1853:,
1795:.
1552:.
387:+
313:.)
299:.)
232:A
188:.
158:)
120:ab
113:×
67:,
2910:)
2906:(
2896:e
2889:t
2882:v
2832:.
2771:.
2740:.
2583:.
2580:s
2578:A
2573:q
2571:A
2566:l
2564:A
2559:s
2557:A
2552:q
2550:A
2545:l
2543:A
2538:q
2536:A
2464:L
2444:G
2440:a
2436:A
2372:n
2366:n
2360:n
2271:E
2267:E
2231:.
2228:E
2225:D
2222:=
2219:E
2216:B
2195:E
2192:C
2189:=
2186:E
2183:A
2128:.
2113:E
2110:C
2107:D
2098:E
2095:A
2092:B
2065:E
2062:D
2059:C
2050:E
2047:B
2044:A
1995:.
1991:|
1984:1
1977:2
1973:c
1965:1
1961:c
1953:1
1946:2
1942:b
1934:1
1930:b
1922:1
1915:2
1911:a
1903:1
1899:a
1891:|
1887:=
1884:K
1871:c
1867:b
1863:a
1859:c
1855:b
1851:a
1831:2
1826:R
1818:c
1815:,
1812:b
1809:,
1806:a
1781:)
1775:T
1770:V
1766:V
1763:(
1748:b
1744:a
1728:n
1722:2
1717:R
1707:]
1699:n
1695:b
1682:2
1678:b
1670:1
1666:b
1656:n
1652:a
1639:2
1635:a
1627:1
1623:a
1616:[
1611:=
1608:V
1586:n
1581:R
1572:b
1568:,
1564:a
1538:|
1532:1
1528:b
1522:2
1518:a
1509:2
1505:b
1499:1
1495:a
1490:|
1486:=
1482:|
1478:)
1475:V
1472:(
1465:|
1454:b
1450:a
1446:b
1442:a
1426:2
1420:2
1415:R
1405:]
1397:2
1393:b
1385:1
1381:b
1371:2
1367:a
1359:1
1355:a
1348:[
1343:=
1340:V
1318:2
1313:R
1304:b
1300:,
1296:a
1263:2
1259:/
1255:)
1250:1
1246:D
1242:+
1239:C
1236:+
1233:B
1230:(
1227:=
1224:S
1201:,
1196:)
1191:1
1187:D
1180:C
1177:+
1174:B
1171:(
1168:)
1163:1
1159:D
1155:+
1152:C
1146:B
1143:(
1140:)
1135:1
1131:D
1127:+
1124:C
1121:+
1118:B
1112:(
1109:)
1104:1
1100:D
1096:+
1093:C
1090:+
1087:B
1084:(
1077:2
1074:1
1069:=
1064:)
1059:1
1055:D
1048:S
1045:(
1042:)
1039:C
1033:S
1030:(
1027:)
1024:B
1018:S
1015:(
1012:S
1007:2
1004:=
1001:K
984:1
981:D
977:C
973:B
956:.
952:|
946:2
942:C
933:2
929:B
924:|
915:2
910:|
896:|
889:=
886:K
853:C
849:B
831:.
816:C
810:B
807:=
804:K
791:C
787:B
770:.
767:H
761:B
758:=
755:)
752:H
746:A
743:(
737:)
734:H
728:)
725:A
722:+
719:B
716:(
713:(
710:=
701:K
694:2
682:K
678:=
675:K
648:.
645:H
637:2
634:A
629:=
620:K
592:H
586:)
583:A
580:+
577:B
574:(
571:=
562:K
548:K
523:.
520:h
517:b
514:=
511:K
482:h
478:b
465:.
453:h
450:b
447:=
444:K
397:b
393:a
389:b
385:a
380:.
322:X
318:X
288:.
278:.
124:θ
115:h
111:b
97:2
95:C
84:4
34:.
20:)
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