81:
1585:
1515:
1503:
1298:, meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order. A great example of perpendicularity can be seen in any compass, note the cardinal points; North, East, South, West (NESW) The line N-S is perpendicular to the line W-E and the angles N-E, E-S, S-W and W-N are all 90Ā° to one another.
1831:
38:
1575:
can be used as the basis of methods of constructing right angles. For example, by counting links, three pieces of chain can be made with lengths in the ratio 3:4:5. These can be laid out to form a triangle, which will have a right angle opposite its longest side. This method is useful for laying out
2904:
of a parabola is that If two tangents to the parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular. This implies that, seen from any point on its directrix, any parabola subtends a right angle.
2846:
states that two lines both through the same point on a circle but going through opposite endpoints of a diameter are perpendicular. This is equivalent to saying that any diameter of a circle subtends a right angle at any point on the circle, except the two endpoints of the diameter.
1273:; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its
2472:
2325:
2596:
1282:
A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the
1745:
is measured as the length from the point along a segment that is perpendicular to the plane, meaning that it is perpendicular to all lines in the plane that pass through the nearest point in the plane to the given point.
1449:
is frequently used in connection with perpendiculars. This usage is exemplified in the top diagram, above, and its caption. The diagram can be in any orientation. The foot is not necessarily at the bottom.
2824:
The sum of the squared lengths of any two perpendicular chords intersecting at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8
1421:
2750:
2646:
2039:
1974:
1371:
1339:
3118:, if squares are constructed externally on the sides of a quadrilateral, the line segments connecting the centers of opposite squares are perpendicular and equal in length.
2135:
2085:
1520:
Construction of the perpendicular to the half-line h from the point P (applicable not only at the end point A, M is freely selectable), animation at the end with pause 10 s
2964:
2724:
2924:
The product of the perpendicular distances from a point P on a hyperbola or on its conjugate hyperbola to the asymptotes is a constant independent of the location of P.
2331:
2184:
2176:
41:
The segment AB is perpendicular to the segment CD because the two angles it creates (indicated in orange and blue) are each 90 degrees. The segment AB can be called
2700:
2673:
1634:
In the figure at the right, all of the orange-shaded angles are congruent to each other and all of the green-shaded angles are congruent to each other, because
2886:, the axis of symmetry is perpendicular to each of the latus rectum, the directrix, and the tangent line at the point where the axis intersects the parabola.
2478:
1209:
1430:
if it is perpendicular to every line in the plane that it intersects. This definition depends on the definition of perpendicularity between lines.
1373:
if, when each is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols,
1546:
Step 2 (green): construct circles centered at A' and B' having equal radius. Let Q and P be the points of intersection of these two circles.
1528:
309:
1576:
gardens and fields, where the dimensions are large, and great accuracy is not needed. The chains can be used repeatedly whenever required.
1264:, ā. Perpendicular intersections can happen between two lines (or two line segments), between a line and a plane, and between two planes.
1762:
3111:, a line through the midpoint of one side and through the intersection point of the diagonals is perpendicular to the opposite side.
275:
3156:
1754:
1742:
1631:. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.
1716:
1638:
are congruent and alternate interior angles formed by a transversal cutting parallel lines are congruent. Therefore, if lines
1202:
1156:
762:
221:
31:
2863:
are perpendicular to each other and to the tangent lines to the ellipse at the points where the axes intersect the ellipse.
1773:
fits a line to data points by minimizing the sum of squared perpendicular distances from the data points to the line. Other
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so that the origin is situated where the lines cross. Then define two displacements along each line,
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113:
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2706:
Both proofs are valid for horizontal and vertical lines to the extent that we can let one slope be
2467:{\displaystyle {\vec {r}}_{2}=x_{2}{\hat {x}}+y_{2}{\hat {y}}=x_{2}{\hat {x}}+m_{2}x_{2}{\hat {y}}}
2320:{\displaystyle {\vec {r}}_{1}=x_{1}{\hat {x}}+y_{1}{\hat {y}}=x_{1}{\hat {x}}+m_{1}x_{1}{\hat {y}}}
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1766:, for the perpendicular distance between two non-parallel lines in three-dimensional space
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174:
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In the two-dimensional plane, right angles can be formed by two intersected lines if the
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867:
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747:
2591:{\displaystyle {\vec {r}}_{1}\cdot {\vec {r}}_{2}=\left(1+m_{1}m_{2}\right)x_{1}x_{2}=0}
1267:
Perpendicularity is one particular instance of the more general mathematical concept of
3250:
College
Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle
2979:
1816:
1804:
1434:
1306:
1284:
1172:
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1080:
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1758:, for the perpendicular distance from the origin to a plane in three-dimensional space
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792:
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How to draw a perpendicular at the endpoint of a ray with compass and straight edge
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2787:
If the intersection of any two perpendicular chords divides one chord into lengths
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1735:
1724:
1302:
1235:
1090:
1039:
852:
707:
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412:
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1777:
methods using perpendicular distance to measure the quality of a fit exist, as in
1619:), all of the angles formed along the third line are right angles. Therefore, in
1557:
for QPA' and QPB' to conclude that angles OPA' and OPB' are equal. Then use the
3285:
3020:
2090:
1540:
1239:
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999:
817:
752:
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652:
627:
17:
1653:
One of the orange-shaded angles is congruent to one of the green-shaded angles.
3005:
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732:
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483:
3292:
How to draw a perpendicular bisector of a line with compass and straight edge
2178:
Now, use the fact that the inner product vanishes for perpendicular vectors:
3069:
3062:
3054:
3019:
concerns a property of two perpendicular lines intersecting at a triangle's
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2918:
2914:
1508:
Construction of the perpendicular (blue) to the line AB through the point P.
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994:
712:
675:
539:
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1646:
are parallel, any of the following conclusions leads to all of the others:
1561:
for triangles OPA' and OPB' to conclude that angles POA and POB are equal.
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3077:
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2990:
2901:
2890:
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1705:
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1549:
Step 3 (blue): connect Q and P to construct the desired perpendicular PQ.
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To make the perpendicular to the line g at or through the point P using
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to that circle at the point where the diameter intersects the circle.
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with center at P to create points A' and B' on the line AB, which are
27:
Relationship between two lines that meet at a right angle (90 degrees)
3092:
3050:
2840:
is the distance from the center point to the point of intersection).
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is perpendicular to the line from that point through the parabola's
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To make the perpendicular to the line AB through the point P using
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on one side of the first line is cut by the second line into two
3272:
1623:, any two lines that are both perpendicular to a third line are
605:
455:
3001:
of the sides also play a prominent role in triangle geometry.
2921:
is perpendicular to the conjugate axis and to each directrix.
3065:
that has two pairs of adjacent sides that are perpendicular.
2889:
From a point on the tangent line to a parabola's vertex, the
1784:
The concept of perpendicular distance may be generalized to
1423:
means line segment AB is perpendicular to line segment CD.
1788:
orthogonal distance, between more abstract non-geometric
1734:
is measured by a line segment that is perpendicular to a
1727:
from it to the given point is perpendicular to the line.
1433:
Two planes in space are said to be perpendicular if the
2752:
If one slope goes to zero, the other goes to infinity.
2041:, the graphs of the functions will be perpendicular if
1416:{\displaystyle {\overline {AB}}\perp {\overline {CD}}}
3130:
can be pairwise perpendicular, as exemplified by the
3029:
concerns the relationship of line segments through a
2945:
2866:
The major axis of an ellipse is perpendicular to the
2780:
A line segment through a circle's center bisecting a
2732:
2712:
2681:
2654:
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2187:
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2107:
2047:
1982:
1917:
1553:
To prove that the PQ is perpendicular to AB, use the
1379:
1347:
1315:
3057:, all pairs of adjacent sides are perpendicular. A
2097:can be also used to obtain the same result: First,
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2744:
2718:
2694:
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2590:
2466:
2319:
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2129:
2079:
2033:
1968:
1650:One of the angles in the diagram is a right angle.
1415:
1365:
1333:
3107:, in an orthodiagonal quadrilateral that is also
2935:that are perpendicular to each other. It has an
1738:to the curve at the nearest point on the curve.
1246:that are 90 degrees or Ļ/2 radians wide) at the
3215:, Dover, 2nd edition, 1996: pp. 104ā105, #4ā23.
1203:
45:, using "perpendicular" as a noun. The point
8:
3254:(2nd ed.), New York: Barnes & Noble
1588:The arrowhead marks indicate that the lines
3229:29(4), September 1998, p. 331, problem 635.
1723:on that line. That is the point at which a
2745:{\displaystyle \varepsilon \rightarrow 0.}
1615:) are both perpendicular to a third line (
1210:
1196:
925:
444:
79:
68:
43:the perpendicular from A to the segment CD
3076:is a perpendicular to a side through the
3012:is perpendicular to the triangle's base.
2946:
2944:
2795:and divides the other chord into lengths
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1730:Likewise, the distance from a point to a
1398:
1380:
1378:
1348:
1346:
1316:
1314:
1258:may be represented graphically using the
2641:{\displaystyle \therefore m_{1}m_{2}=-1}
1708:from one to the other, measured along a
1583:
1531:, proceed as follows (see figure left):
1426:A line is said to be perpendicular to a
3168:
1164:
1098:
1047:
976:
928:
690:
552:
529:
496:
468:
71:
2993:are perpendicular to their respective
1712:that is perpendicular to one or both.
1294:. Perpendicularity can be shown to be
310:Straightedge and compass constructions
3091:are perpendicular. These include the
2034:{\displaystyle y_{2}(x)=m_{2}x+b_{2}}
1969:{\displaystyle y_{1}(x)=m_{1}x+b_{1}}
1807:, between an arbitrary point and its
1529:compass-and-straightedge construction
1437:at which they meet is a right angle.
7:
1834:Two perpendicular lines have slopes
3199:
3187:
3175:
2821:equals the square of the diameter.
1811:on the surface. It can be used for
1763:Nearest distance between skew lines
1341:is perpendicular to a line segment
1301:Perpendicularity easily extends to
2891:other tangent line to the parabola
25:
2982:are perpendicular to each other.
1580:In relationship to parallel lines
1493:Construction of the perpendicular
276:Noncommutative algebraic geometry
3213:Challenging Problems in Geometry
3157:Tangential and normal components
1755:Point on plane closest to origin
1743:distance from a point to a plane
1688:This section is an excerpt from
1513:
1501:
1465:is the point of intersection of
1366:{\displaystyle {\overline {CD}}}
1334:{\displaystyle {\overline {AB}}}
2784:is perpendicular to the chord.
1719:is the distance to the nearest
1717:distance from a point to a line
51:foot of the perpendicular from
3033:and perpendicular to any line
2736:
2511:
2489:
2458:
2423:
2398:
2373:
2342:
2311:
2276:
2251:
2226:
2195:
2162:
2144:
2130:{\displaystyle {\vec {r}}_{j}}
2115:
2080:{\displaystyle m_{1}m_{2}=-1.}
1999:
1993:
1934:
1928:
1627:to each other, because of the
1568:, see the animation at right.
1485:of this perpendicular through
1309:. For example, a line segment
669:- / other-dimensional
32:Perpendicular (disambiguation)
1:
1880:satisfying the relationship
1803:normal distance, involving a
1798:principal components analysis
3138:axes of a three-dimensional
2959:{\displaystyle {\sqrt {2}}.}
2719:{\displaystyle \varepsilon }
1469:and the unique line through
1408:
1390:
1358:
1326:
3288:with interactive animation.
3226:College Mathematics Journal
3140:Cartesian coordinate system
3085:orthodiagonal quadrilateral
2836:is the circle's radius and
2756:In circles and other conics
1704:between two objects is the
3330:
3265:Holt, Rinehart and Winston
2726:, and take the limit that
1687:
1678:All four angles are equal.
1535:Step 1 (red): construct a
29:
3300:(animated demonstration).
3294:(animated demonstration).
3286:Definition: perpendicular
3211:Posamentier and Salkind,
3122:Lines in three dimensions
3087:is a quadrilateral whose
1908:equals ā1. Thus for two
1749:Other instances include:
1671:is perpendicular to line
1660:is perpendicular to line
1473:that is perpendicular to
1250:of intersection called a
3244:Altshiller-Court, Nathan
2773:is perpendicular to the
2171:{\displaystyle (j=1,2).}
1771:Perpendicular regression
165:Non-Archimedean geometry
3128:three-dimensional space
3017:Droz-Farny line theorem
2999:perpendicular bisectors
1775:geometric curve fitting
1441:Foot of a perpendicular
271:Noncommutative geometry
3259:Kay, David C. (1969),
3080:of the opposite side.
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2720:
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2321:
2172:
2131:
2081:
2035:
1970:
1897:
1702:perpendicular distance
1690:Perpendicular distance
1683:In computing distances
1604:
1559:SAS congruence theorem
1555:SSS congruence theorem
1417:
1367:
1335:
239:Discrete/Combinatorial
66:
3152:Orthogonal projection
3126:Up to three lines in
3105:Brahmagupta's theorem
2961:
2929:rectangular hyperbola
2747:
2721:
2697:
2695:{\displaystyle x_{2}}
2670:
2668:{\displaystyle x_{1}}
2643:
2593:
2469:
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2173:
2132:
2082:
2036:
1971:
1833:
1587:
1418:
1368:
1336:
222:Discrete differential
40:
2943:
2855:The major and minor
2730:
2710:
2679:
2652:
2603:
2479:
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2185:
2141:
2105:
2045:
1980:
1915:
1453:More precisely, let
1377:
1345:
1313:
1261:perpendicular symbol
30:For other uses, see
3116:van Aubel's theorem
1779:total least squares
1573:Pythagorean theorem
1254:. The condition of
489:Pythagorean theorem
3037:to the triangle's
3027:Harcourt's theorem
3010:isosceles triangle
2956:
2902:orthoptic property
2742:
2716:
2692:
2665:
2638:
2588:
2464:
2317:
2168:
2127:
2077:
2031:
1966:
1898:
1826:Graph of functions
1629:parallel postulate
1621:Euclidean geometry
1605:
1413:
1363:
1331:
67:
3068:Each of the four
2951:
2514:
2492:
2461:
2426:
2401:
2376:
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2279:
2254:
2229:
2198:
2118:
2099:shift coordinates
1815:and for defining
1411:
1393:
1361:
1329:
1228:geometric objects
1220:
1219:
1185:
1184:
908:List of geometers
591:Three-dimensional
580:
579:
57:, or simply, the
16:(Redirected from
3321:
3275:
3261:College Geometry
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1598:transversal line
1566:Thales's theorem
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1256:perpendicularity
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378:Zero-dimensional
83:
69:
21:
18:Perpendicularity
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3059:right trapezoid
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2915:transverse axis
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2853:
2844:Thales' theorem
2804:
2763:
2758:
2728:
2727:
2708:
2707:
2682:
2677:
2676:
2655:
2650:
2649:
2619:
2609:
2601:
2600:
2572:
2562:
2547:
2537:
2530:
2526:
2504:
2482:
2477:
2476:
2442:
2432:
2407:
2382:
2357:
2335:
2330:
2329:
2295:
2285:
2260:
2235:
2210:
2188:
2183:
2182:
2139:
2138:
2108:
2103:
2102:
2058:
2048:
2043:
2042:
2021:
2005:
1983:
1978:
1977:
1956:
1940:
1918:
1913:
1912:
1893:
1887:
1881:
1878:
1871:
1864:
1858:
1855:
1848:
1841:
1835:
1828:
1823:
1822:
1817:offset surfaces
1813:surface fitting
1792:objects, as in
1693:
1685:
1636:vertical angles
1603:, are parallel.
1582:
1525:
1524:
1523:
1522:
1521:
1518:
1510:
1509:
1506:
1495:
1486:
1478:
1474:
1470:
1466:
1462:
1458:
1457:be a point and
1454:
1443:
1400:
1382:
1375:
1374:
1350:
1343:
1342:
1318:
1311:
1310:
1216:
1187:
1186:
923:
922:
913:
912:
703:
702:
686:
685:
671:
670:
658:
657:
594:
593:
582:
581:
442:
441:
439:Two-dimensional
430:
429:
403:
402:
400:One-dimensional
391:
390:
381:
380:
369:
368:
302:
301:
300:
283:
282:
131:
130:
119:
96:
35:
28:
23:
22:
15:
12:
11:
5:
3327:
3325:
3317:
3316:
3306:
3305:
3302:
3301:
3295:
3289:
3281:
3280:External links
3278:
3277:
3276:
3256:
3238:
3235:
3232:
3231:
3217:
3204:
3192:
3180:
3178:, p. 114)
3167:
3166:
3164:
3161:
3160:
3159:
3154:
3147:
3144:
3123:
3120:
3046:
3045:Quadrilaterals
3043:
2980:right triangle
2978:The legs of a
2975:
2972:
2970:
2967:
2955:
2950:
2910:
2907:
2879:
2876:
2852:
2849:
2762:
2759:
2757:
2754:
2741:
2738:
2735:
2715:
2704:
2703:
2689:
2685:
2662:
2658:
2637:
2634:
2631:
2626:
2622:
2616:
2612:
2608:
2598:
2587:
2584:
2579:
2575:
2569:
2565:
2560:
2554:
2550:
2544:
2540:
2536:
2533:
2529:
2525:
2520:
2513:
2510:
2503:
2498:
2491:
2488:
2474:
2460:
2457:
2449:
2445:
2439:
2435:
2431:
2425:
2422:
2414:
2410:
2406:
2400:
2397:
2389:
2385:
2381:
2375:
2372:
2364:
2360:
2356:
2351:
2344:
2341:
2327:
2313:
2310:
2302:
2298:
2292:
2288:
2284:
2278:
2275:
2267:
2263:
2259:
2253:
2250:
2242:
2238:
2234:
2228:
2225:
2217:
2213:
2209:
2204:
2197:
2194:
2167:
2164:
2161:
2158:
2155:
2152:
2149:
2146:
2124:
2117:
2114:
2076:
2073:
2070:
2065:
2061:
2055:
2051:
2028:
2024:
2020:
2017:
2012:
2008:
2004:
2001:
1998:
1995:
1990:
1986:
1963:
1959:
1955:
1952:
1947:
1943:
1939:
1936:
1933:
1930:
1925:
1921:
1891:
1885:
1876:
1869:
1862:
1853:
1846:
1839:
1827:
1824:
1821:
1820:
1805:surface normal
1801:
1794:linear algebra
1768:
1767:
1759:
1694:
1686:
1684:
1681:
1680:
1679:
1676:
1665:
1654:
1651:
1607:If two lines (
1581:
1578:
1551:
1550:
1547:
1544:
1519:
1512:
1511:
1507:
1500:
1499:
1498:
1497:
1496:
1494:
1491:
1481:is called the
1442:
1439:
1435:dihedral angle
1410:
1406:
1403:
1397:
1392:
1388:
1385:
1360:
1356:
1353:
1328:
1324:
1321:
1285:straight angle
1218:
1217:
1215:
1214:
1207:
1200:
1192:
1189:
1188:
1183:
1182:
1181:
1180:
1175:
1167:
1166:
1162:
1161:
1160:
1159:
1154:
1149:
1144:
1139:
1134:
1129:
1124:
1119:
1114:
1109:
1101:
1100:
1096:
1095:
1094:
1093:
1088:
1083:
1078:
1073:
1068:
1063:
1058:
1050:
1049:
1045:
1044:
1043:
1042:
1037:
1032:
1027:
1022:
1017:
1012:
1007:
1002:
997:
992:
987:
979:
978:
974:
973:
972:
971:
966:
961:
956:
951:
946:
941:
933:
932:
924:
920:
919:
918:
915:
914:
911:
910:
905:
900:
895:
890:
885:
880:
875:
870:
865:
860:
855:
850:
845:
840:
835:
830:
825:
820:
815:
810:
805:
800:
795:
790:
785:
780:
775:
770:
765:
760:
755:
750:
745:
740:
735:
730:
725:
720:
715:
710:
704:
700:
699:
698:
695:
694:
688:
687:
684:
683:
678:
672:
665:
664:
663:
660:
659:
656:
655:
650:
645:
643:Platonic Solid
640:
635:
630:
625:
620:
615:
614:
613:
602:
601:
595:
589:
588:
587:
584:
583:
578:
577:
576:
575:
570:
565:
557:
556:
550:
549:
548:
547:
542:
534:
533:
527:
526:
525:
524:
519:
514:
509:
501:
500:
494:
493:
492:
491:
486:
481:
473:
472:
466:
465:
464:
463:
458:
453:
443:
437:
436:
435:
432:
431:
428:
427:
422:
421:
420:
415:
404:
398:
397:
396:
393:
392:
389:
388:
382:
376:
375:
374:
371:
370:
367:
366:
361:
356:
350:
349:
344:
339:
329:
324:
319:
313:
312:
303:
299:
298:
295:
291:
290:
289:
288:
285:
284:
281:
280:
279:
278:
268:
263:
258:
253:
248:
247:
246:
236:
231:
226:
225:
224:
219:
214:
204:
203:
202:
197:
187:
182:
177:
172:
167:
162:
161:
160:
155:
154:
153:
138:
132:
126:
125:
124:
121:
120:
118:
117:
107:
101:
98:
97:
84:
76:
75:
49:is called the
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3326:
3315:
3314:Orthogonality
3312:
3311:
3309:
3299:
3296:
3293:
3290:
3287:
3284:
3283:
3279:
3274:
3270:
3266:
3262:
3257:
3252:
3251:
3245:
3241:
3240:
3236:
3228:
3227:
3221:
3218:
3214:
3208:
3205:
3202:, p. 91)
3201:
3196:
3193:
3190:, p. 91)
3189:
3184:
3181:
3177:
3172:
3169:
3162:
3158:
3155:
3153:
3150:
3149:
3145:
3143:
3141:
3137:
3133:
3129:
3121:
3119:
3117:
3112:
3110:
3106:
3102:
3098:
3094:
3090:
3086:
3081:
3079:
3075:
3074:quadrilateral
3071:
3066:
3064:
3060:
3056:
3052:
3044:
3042:
3040:
3036:
3032:
3028:
3024:
3022:
3018:
3013:
3011:
3007:
3002:
3000:
2996:
2992:
2988:
2983:
2981:
2973:
2968:
2966:
2953:
2948:
2938:
2934:
2930:
2925:
2922:
2920:
2916:
2908:
2906:
2903:
2898:
2896:
2892:
2887:
2885:
2877:
2875:
2873:
2869:
2864:
2862:
2858:
2850:
2848:
2845:
2841:
2839:
2835:
2831:
2827:
2822:
2819:
2815:
2811:
2807:
2802:
2798:
2794:
2790:
2785:
2783:
2778:
2776:
2772:
2768:
2760:
2755:
2753:
2739:
2733:
2713:
2687:
2683:
2660:
2656:
2635:
2632:
2629:
2624:
2620:
2614:
2610:
2606:
2599:
2585:
2582:
2577:
2573:
2567:
2563:
2558:
2552:
2548:
2542:
2538:
2534:
2531:
2527:
2523:
2518:
2508:
2501:
2496:
2486:
2475:
2455:
2447:
2443:
2437:
2433:
2429:
2420:
2412:
2408:
2404:
2395:
2387:
2383:
2379:
2370:
2362:
2358:
2354:
2349:
2339:
2328:
2308:
2300:
2296:
2290:
2286:
2282:
2273:
2265:
2261:
2257:
2248:
2240:
2236:
2232:
2223:
2215:
2211:
2207:
2202:
2192:
2181:
2180:
2179:
2165:
2159:
2156:
2153:
2150:
2147:
2122:
2112:
2100:
2096:
2092:
2087:
2074:
2071:
2068:
2063:
2059:
2053:
2049:
2026:
2022:
2018:
2015:
2010:
2006:
2002:
1996:
1988:
1984:
1961:
1957:
1953:
1950:
1945:
1941:
1937:
1931:
1923:
1919:
1911:
1907:
1903:
1890:
1884:
1875:
1868:
1861:
1852:
1845:
1838:
1832:
1825:
1818:
1814:
1810:
1806:
1802:
1799:
1795:
1791:
1787:
1786:
1785:
1782:
1780:
1776:
1772:
1765:
1764:
1760:
1757:
1756:
1752:
1751:
1750:
1747:
1744:
1739:
1737:
1733:
1728:
1726:
1722:
1718:
1713:
1711:
1707:
1703:
1699:
1691:
1682:
1677:
1674:
1670:
1666:
1663:
1659:
1655:
1652:
1649:
1648:
1647:
1645:
1641:
1637:
1632:
1630:
1626:
1622:
1618:
1614:
1610:
1602:
1599:
1596:, cut by the
1595:
1591:
1586:
1579:
1577:
1574:
1569:
1567:
1562:
1560:
1556:
1548:
1545:
1542:
1538:
1534:
1533:
1532:
1530:
1516:
1504:
1492:
1490:
1484:
1451:
1448:
1440:
1438:
1436:
1431:
1429:
1424:
1404:
1401:
1395:
1386:
1383:
1354:
1351:
1322:
1319:
1308:
1304:
1299:
1297:
1293:
1290:
1286:
1280:
1278:
1277:
1276:normal vector
1272:
1271:
1270:orthogonality
1265:
1263:
1262:
1257:
1253:
1249:
1245:
1241:
1237:
1233:
1232:perpendicular
1229:
1225:
1213:
1208:
1206:
1201:
1199:
1194:
1193:
1191:
1190:
1179:
1176:
1174:
1171:
1170:
1169:
1168:
1163:
1158:
1155:
1153:
1150:
1148:
1145:
1143:
1140:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1104:
1103:
1102:
1097:
1092:
1089:
1087:
1084:
1082:
1079:
1077:
1074:
1072:
1069:
1067:
1064:
1062:
1059:
1057:
1054:
1053:
1052:
1051:
1046:
1041:
1038:
1036:
1033:
1031:
1028:
1026:
1023:
1021:
1018:
1016:
1013:
1011:
1008:
1006:
1003:
1001:
998:
996:
993:
991:
988:
986:
983:
982:
981:
980:
975:
970:
967:
965:
962:
960:
957:
955:
952:
950:
947:
945:
942:
940:
937:
936:
935:
934:
931:
927:
917:
916:
909:
906:
904:
901:
899:
896:
894:
891:
889:
886:
884:
881:
879:
876:
874:
871:
869:
866:
864:
861:
859:
856:
854:
851:
849:
846:
844:
841:
839:
836:
834:
831:
829:
826:
824:
821:
819:
816:
814:
811:
809:
806:
804:
801:
799:
796:
794:
791:
789:
786:
784:
781:
779:
776:
774:
771:
769:
766:
764:
761:
759:
756:
754:
751:
749:
746:
744:
741:
739:
736:
734:
731:
729:
726:
724:
721:
719:
716:
714:
711:
709:
706:
705:
697:
696:
693:
689:
682:
679:
677:
674:
673:
668:
662:
661:
654:
651:
649:
646:
644:
641:
639:
636:
634:
631:
629:
626:
624:
621:
619:
616:
612:
609:
608:
607:
604:
603:
600:
597:
596:
592:
586:
585:
574:
571:
569:
568:Circumference
566:
564:
561:
560:
559:
558:
555:
551:
546:
543:
541:
538:
537:
536:
535:
532:
531:Quadrilateral
528:
523:
520:
518:
515:
513:
510:
508:
505:
504:
503:
502:
499:
498:Parallelogram
495:
490:
487:
485:
482:
480:
477:
476:
475:
474:
471:
467:
462:
459:
457:
454:
452:
449:
448:
447:
446:
440:
434:
433:
426:
423:
419:
416:
414:
411:
410:
409:
406:
405:
401:
395:
394:
387:
384:
383:
379:
373:
372:
365:
362:
360:
357:
355:
352:
351:
348:
345:
343:
340:
337:
336:Perpendicular
333:
332:Orthogonality
330:
328:
325:
323:
320:
318:
315:
314:
311:
308:
307:
306:
296:
293:
292:
287:
286:
277:
274:
273:
272:
269:
267:
264:
262:
259:
257:
256:Computational
254:
252:
249:
245:
242:
241:
240:
237:
235:
232:
230:
227:
223:
220:
218:
215:
213:
210:
209:
208:
205:
201:
198:
196:
193:
192:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
159:
156:
152:
149:
148:
147:
144:
143:
142:
141:Non-Euclidean
139:
137:
134:
133:
129:
123:
122:
115:
111:
108:
106:
103:
102:
100:
99:
95:
91:
87:
82:
78:
77:
74:
70:
64:
60:
56:
55:to segment CD
52:
48:
44:
39:
33:
19:
3263:, New York:
3260:
3249:
3224:
3220:
3212:
3207:
3195:
3183:
3171:
3135:
3131:
3125:
3113:
3082:
3067:
3048:
3025:
3014:
3003:
2984:
2977:
2937:eccentricity
2926:
2923:
2912:
2899:
2888:
2881:
2872:latus rectum
2870:and to each
2865:
2854:
2842:
2837:
2833:
2829:
2825:
2823:
2817:
2813:
2809:
2805:
2800:
2796:
2792:
2788:
2786:
2779:
2775:tangent line
2764:
2705:
2088:
1899:
1888:
1882:
1873:
1866:
1859:
1850:
1843:
1836:
1783:
1769:
1761:
1753:
1748:
1740:
1736:tangent line
1729:
1714:
1695:
1672:
1668:
1661:
1657:
1643:
1639:
1633:
1616:
1612:
1608:
1606:
1600:
1593:
1589:
1570:
1563:
1552:
1526:
1482:
1452:
1446:
1444:
1432:
1425:
1300:
1281:
1274:
1268:
1266:
1259:
1255:
1251:
1240:right angles
1236:intersection
1231:
1221:
1040:Parameshvara
853:Parameshvara
623:Dodecahedron
335:
207:Differential
62:
58:
54:
50:
46:
42:
3021:orthocenter
2969:In polygons
2091:dot product
1541:equidistant
1461:a line. If
1165:Present day
1112:Lobachevsky
1099:1700sā1900s
1056:Jyeį¹£į¹hadeva
1048:1400sā1700s
1000:Brahmagupta
823:Lobachevsky
803:Jyeį¹£į¹hadeva
753:Brahmagupta
681:Hypersphere
653:Tetrahedron
628:Icosahedron
200:Diophantine
3237:References
3099:, and the
3070:maltitudes
3006:Euler line
2933:asymptotes
2909:Hyperbolas
2702:vanishes.)
1894:= −1
1790:orthogonal
1025:al-Yasamin
969:Apollonius
964:Archimedes
954:Pythagoras
944:Baudhayana
898:al-Yasamin
848:Pythagoras
743:Baudhayana
733:Archimedes
728:Apollonius
633:Octahedron
484:Hypotenuse
359:Similarity
354:Congruence
266:Incidence
217:Symplectic
212:Riemannian
195:Arithmetic
170:Projective
158:Hyperbolic
86:Projecting
3246:(1952) ,
3200:Kay (1969
3188:Kay (1969
3176:Kay (1969
3089:diagonals
3063:trapezoid
3055:rectangle
3053:or other
2987:altitudes
2974:Triangles
2939:equal to
2919:hyperbola
2878:Parabolas
2868:directrix
2737:→
2734:ε
2714:ε
2648:(unless
2633:−
2607:∴
2512:→
2502:⋅
2490:→
2459:^
2424:^
2399:^
2374:^
2343:→
2312:^
2277:^
2252:^
2227:^
2196:→
2116:→
2072:−
1904:of their
1445:The word
1409:¯
1396:⊥
1391:¯
1359:¯
1327:¯
1296:symmetric
1289:congruent
1234:if their
1142:Minkowski
1061:Descartes
995:Aryabhata
990:KÄtyÄyana
921:by period
833:Minkowski
808:KÄtyÄyana
768:Descartes
713:Aryabhata
692:Geometers
676:Tesseract
540:Trapezoid
512:Rectangle
305:Dimension
190:Algebraic
180:Synthetic
151:Spherical
136:Euclidean
3308:Category
3273:69-12075
3146:See also
3078:midpoint
3039:incircle
2991:triangle
2884:parabola
2851:Ellipses
2767:diameter
1706:distance
1698:geometry
1625:parallel
1303:segments
1224:geometry
1132:PoincarƩ
1076:Minggatu
1035:Yang Hui
1005:Virasena
893:Yang Hui
888:Virasena
858:PoincarƩ
838:Minggatu
618:Cylinder
563:Diameter
522:Rhomboid
479:Altitude
470:Triangle
364:Symmetry
342:Parallel
327:Diagonal
297:Features
294:Concepts
185:Analytic
146:Elliptic
128:Branches
114:Timeline
73:Geometry
59:foot of
3097:rhombus
3035:tangent
2861:ellipse
2832:(where
2803:, then
2761:Circles
2095:vectors
1902:product
1796:(e.g.,
1725:segment
1543:from P.
1477:, then
1157:Coxeter
1137:Hilbert
1122:Riemann
1071:Huygens
1030:al-Tusi
1020:KhayyƔm
1010:Alhazen
977:1ā1400s
878:al-Tusi
863:Riemann
813:KhayyƔm
798:Huygens
793:Hilbert
763:Coxeter
723:Alhazen
701:by name
638:Pyramid
517:Rhombus
461:Polygon
413:segment
261:Fractal
244:Digital
229:Complex
110:History
105:Outline
3271:
3134:, and
3109:cyclic
3095:, the
3093:square
3051:square
3031:vertex
3008:of an
2997:. The
2859:of an
2771:circle
2137:, for
1906:slopes
1700:, the
1537:circle
1292:angles
1244:angles
1238:forms
1226:, two
1178:Gromov
1173:Atiyah
1152:Veblen
1147:Cartan
1117:Bolyai
1086:Sakabe
1066:Pascal
959:Euclid
949:Manava
883:Veblen
868:Sakabe
843:Pascal
828:Manava
788:Gromov
773:Euclid
758:Cartan
748:Bolyai
738:Atiyah
648:Sphere
611:cuboid
599:Volume
554:Circle
507:Square
425:Length
347:Vertex
251:Convex
234:Finite
175:Affine
90:sphere
3163:Notes
3103:. By
3072:of a
3061:is a
3049:In a
2995:bases
2989:of a
2917:of a
2895:focus
2882:In a
2782:chord
2769:of a
2765:Each
1857:and
1732:curve
1721:point
1667:Line
1656:Line
1428:plane
1248:point
1127:Klein
1107:Gauss
1081:Euler
1015:Sijzi
985:Zhang
939:Ahmes
903:Zhang
873:Sijzi
818:Klein
783:Gauss
778:Euler
718:Ahmes
451:Plane
386:Point
322:Curve
317:Angle
94:plane
92:to a
63:on CD
3269:LCCN
3132:x, y
3101:kite
3015:The
3004:The
2985:The
2931:has
2913:The
2900:The
2857:axes
2799:and
2791:and
2089:The
1976:and
1809:foot
1741:The
1715:The
1710:line
1642:and
1611:and
1592:and
1571:The
1483:foot
1447:foot
1307:rays
1305:and
1252:foot
1230:are
1091:Aida
708:Aida
667:Four
606:Cube
573:Area
545:Kite
456:Area
408:Line
3114:By
3083:An
2828:ā 4
2675:or
2093:of
1865:= Ī
1842:= Ī
1696:In
1222:In
930:BCE
418:ray
3310::
3267:,
3142:.
3041:.
3023:.
2927:A
2897:.
2874:.
2816:+
2812:+
2808:+
2740:0.
2075:1.
1872:/Ī
1849:/Ī
1800:);
1781:.
1489:.
1279:.
88:a
65:.
3136:z
2954:.
2949:2
2838:p
2834:r
2830:p
2826:r
2818:d
2814:c
2810:b
2806:a
2801:d
2797:c
2793:b
2789:a
2688:2
2684:x
2661:1
2657:x
2636:1
2630:=
2625:2
2621:m
2615:1
2611:m
2586:0
2583:=
2578:2
2574:x
2568:1
2564:x
2559:)
2553:2
2549:m
2543:1
2539:m
2535:+
2532:1
2528:(
2524:=
2519:2
2509:r
2497:1
2487:r
2456:y
2448:2
2444:x
2438:2
2434:m
2430:+
2421:x
2413:2
2409:x
2405:=
2396:y
2388:2
2384:y
2380:+
2371:x
2363:2
2359:x
2355:=
2350:2
2340:r
2309:y
2301:1
2297:x
2291:1
2287:m
2283:+
2274:x
2266:1
2262:x
2258:=
2249:y
2241:1
2237:y
2233:+
2224:x
2216:1
2212:x
2208:=
2203:1
2193:r
2166:.
2163:)
2160:2
2157:,
2154:1
2151:=
2148:j
2145:(
2123:j
2113:r
2069:=
2064:2
2060:m
2054:1
2050:m
2027:2
2023:b
2019:+
2016:x
2011:2
2007:m
2003:=
2000:)
1997:x
1994:(
1989:2
1985:y
1962:1
1958:b
1954:+
1951:x
1946:1
1942:m
1938:=
1935:)
1932:x
1929:(
1924:1
1920:y
1896:.
1892:2
1889:m
1886:1
1883:m
1877:2
1874:x
1870:2
1867:y
1863:2
1860:m
1854:1
1851:x
1847:1
1844:y
1840:1
1837:m
1819:.
1692:.
1675:.
1673:b
1669:c
1664:.
1662:a
1658:c
1644:b
1640:a
1617:c
1613:b
1609:a
1601:c
1594:b
1590:a
1487:A
1479:B
1475:m
1471:A
1467:m
1463:B
1459:m
1455:A
1405:D
1402:C
1387:B
1384:A
1355:D
1352:C
1323:B
1320:A
1242:(
1211:e
1204:t
1197:v
338:)
334:(
116:)
112:(
61:A
53:A
47:B
34:.
20:)
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