407:
1177:
1079:
1872:
1416:
623:
311:
274:- the arbitrary placement of points may be prohibited. In such a paradigm, however, for example, various constructions exist so that arbitrary point placement is unnecessary. It is also worth pointing out that no circle could be constructed without the compass, thus there is no reason in practice for a center point not to exist.
297:
The ability to translate, or copy, a circle to a new center is vital in these proofs and fundamental to establishing the veracity of the theorem. The creation of a new circle with the same radius as the first, but centered at a different point, is the key feature distinguishing the collapsing compass
282:
To prove the above constructions #3 and #4, which are included below, a few necessary intermediary constructions are also explained below since they are used and referenced frequently. These are also compass-only constructions. All constructions below rely on #1,#2,#5, and any other construction that
46:
Though the use of a straightedge can make a construction significantly easier, the theorem shows that any set of points that fully defines a constructed figure can be determined with compass alone, and the only reason to use a straightedge is for the aesthetics of seeing straight lines, which for the
261:
In general constructions there are often several variations that will produce the same result. The choices made in such a variant can be made without loss of generality. However, when a construction is being used to prove that something can be done, it is not necessary to describe all these various
34:
It must be understood that "any geometric construction" refers to figures that contain no straight lines, as it is clearly impossible to draw a straight line without a straightedge. It is understood that a line is determined provided that two distinct points on that line are given or constructed,
2172:
concentric is sufficient, provided that a point on either the centerline through them or the radical axis between them is given, or two parallel lines exist in the plane. A single circle without its center can also be sufficient under the right circumstances. Other unique conditions may exist.
168:
It is understood that a straight line cannot be drawn without a straightedge. A line is considered to be given by any two points, as any such pair define a unique line. In keeping with the intent of the theorem which we aim to prove, the actual line need not be drawn but for aesthetic reasons.
2171:
Additionally, the center itself may be omitted instead of portions of the arc, if it is substituted for something else sufficient, such as a second concentric circle, a second intersecting circle, or a third circle in the plane. Alternatively, a second circle which is neither intersecting nor
2136:
shows that in all the constructions mentioned above, the familiar modern compass with its fixable aperture, which can be used to transfer distances, may be replaced with a "collapsible compass", a compass that collapses whenever it is lifted from a page, so that it may not be directly used to
269:
It is also important to note that some of the constructions below proving the Mohr–Mascheroni theorem require the arbitrary placement of points in space, such as finding the center of a circle when not already provided (see construction below). In some construction paradigms - such as in the
2153:, wherein he proposed that any construction possible by straightedge and compass could be done with straightedge alone. However, the one stipulation is that no less than a single circle with its center identified must be provided. This statement, now known as the
298:
from the modern, rigid compass. With the rigid compass this is a triviality, but with the collapsing compass it is a question of construction possibility. The equivalence of a collapsing compass and a rigid compass was proved by Euclid (Book I Proposition 2 of
2137:
transfer distances. Indeed, Euclid's original constructions use a collapsible compass. It is possible to translate any circle in the plane with a collapsing compass using no more than three additional applications of the compass over that of a rigid compass.
302:) using straightedge and collapsing compass when he, essentially, constructs a copy of a circle with a different center. This equivalence can also be established with (collapsing) compass alone, a proof of which can be found in the main article.
2100:
Thus it has been shown that all of the basic construction one can perform with a straightedge and compass can be done with a compass alone, provided that it is understood that a line cannot be literally drawn but merely defined by two points.
70:
Several proofs of the result are known. Mascheroni's proof of 1797 was generally based on the idea of using reflection in a line as the major tool. Mohr's solution was different. In 1890, August Adler published a proof using the
40:
Any
Euclidean construction, insofar as the given and required elements are points (or circles), may be completed with the compass alone if it can be completed with both the compass and the straightedge
1404:
The compass-only construction of the intersection points of a line and a circle breaks into two cases depending upon whether the center of the circle is or is not collinear with the line.
2129:
and others were able to show in the 16th century that any ruler-and-compass construction could be accomplished with a straightedge and a fixed-width compass (i.e. a rusty compass).
262:
choices and, for the sake of clarity of exposition, only one variant will be given below. However, many constructions come in different forms depending on whether or not they use
116:
752:
871:
2586:
2168:
relaxes the requirement that one full circle be provided, and shows that any small arc of the circle, so long as the center is still provided, is still sufficient.
1659:
1858:
1630:
1561:
818:
795:
843:
772:
689:
143:
need to be proven to be possible by using a compass alone, as these are the foundations of, or elementary steps for, all other constructions. These are:
2556:
2538:
2513:
140:
980:
In the event that the above construction fails (that is, the red circle and the black circle do not intersect in two points), find a point
2591:
2444:
2154:
2126:
2133:
292:
118:. In this way, a stronger version of the theorem was proven in 1990. It also shows the dependence of the theorem on
35:
even though no visual representation of the line will be present. The theorem can be stated more precisely as:
28:
2547:
Posamentier, Alfred S.; Geretschläger, Robert (2016), "8. Mascheroni constructions using only the compass",
406:
263:
87:
72:
59:
in 1672, but his proof languished in obscurity until 1928. The theorem was independently discovered by
2181:
201:
The following notation will be used throughout this article. A circle whose center is located at point
92:
2146:
271:
123:
119:
2196:
2150:
2191:
60:
1176:
2552:
2534:
2509:
2440:
1180:
Compass-only construction of the intersection of two lines (not all construction steps shown)
156:
Creating the one or two points in the intersection of a line and a circle (if they intersect)
2493:
2298:
2229:
2165:
2122:
2118:
1078:
193:
Thus, to prove the theorem, only compass-only constructions for #3 and #4 need to be given.
1871:
1635:
726:
83:
2484:
Hungerbühler, Norbert (1994), "A Short
Elementary Proof of the Mohr–Mascheroni Theorem",
1837:
1609:
1540:
848:
16:
Constructions performed by a compass and straightedge can be performed by a compass alone
2571:
1875:
Compass only construction of intersection of a circle and a line (circle center on line)
800:
777:
720:
if the circles do not intersect in two points see below for an alternative construction.
2413:
are on the circle of inversion and so are invariant under this last unneeded inversion.
2325:
2186:
1415:
828:
757:
159:
Creating the one or two points in the intersection of two circles (if they intersect).
2580:
2158:
668:
2435:
Retz, Merlyn; Keihn, Meta
Darlene (1989), "Compass and Straightedge Constructions",
622:
2497:
2149:
conjectured a variation on the same theme. His work paved the way for the field of
1082:
Compass-only construction of the center of a circle through three points (A, B, C)
1038:
as above (the red and black circles must now intersect in two points). The point
2321:
2259:
Hjelmslev, J. (1928) "Om et af den danske matematiker Georg Mohr udgivet skrift
2114:
177:
This can be done with a compass alone. A straightedge is not required for this.
153:
Creating the point which is the intersection of two existing, non-parallel lines
79:
20:
1571:
If the two circles do not intersect then neither does the circle with the line.
310:
56:
395:(contrary to the assumption), and the two circles are internally tangential.
383:(that is, there is a unique point of intersection of the two circles), then
358:, the other point of intersection of the two circles, is the reflection of
582:
This construction can be repeated as often as necessary to find a point
1667:
An alternate construction, using circle inversion can also be given.
331:
not on the line determined by that segment, construct the image of
2439:, National Council of Teachers of Mathematics (NCTM), p. 195,
2405:
Pedoe carries out one more inversion at this point, but the points
1870:
1414:
1175:
1077:
621:
405:
309:
27:
states that any geometric construction that can be performed by a
410:
A compass-only construction of doubling the length of segment AB
150:
Creating the circle through one point with centre another point
185:
This construction can also be done directly with a compass.
1412:
Assume that center of the circle does not lie on the line.
665:
in the circle. Naturally there is no inversion for a point
710:
Assume that the red circle intersects the black circle at
2326:"On strict strong constructibility with a compass alone"
1172:
Intersection of two non-parallel lines (construction #3)
2533:, Mathematical Association of America, pp. 23–25,
2348:
2346:
1074:
Determining the center of a circle through three points
2529:
Pedoe, Dan (1995) , "1 Section 11: Compass geometry",
671:
47:
purposes of construction is functionally unnecessary.
1911:, the intersection points of the circle and the line.
1840:
1638:
1612:
1543:
1400:
Intersection of a line and a circle (construction #4)
851:
831:
803:
780:
760:
729:
95:
1834:
If the two circles are (internally) tangential then
1537:
If the two circles are (externally) tangential then
560:
is an equilateral triangle, and the three angles at
1690:. We wish to construct the points of intersection,
1443:. We wish to construct the points of intersection,
173:#2 - A circle through one point with defined center
1852:
1653:
1624:
1555:
865:
837:
812:
789:
766:
746:
683:
266:and these alternatives will be given if possible.
110:
287:Compass equivalence theorem (circle translation)
2437:Historical Topics for the Mathematics Classroom
141:basic constructions of compass and straightedge
1019:(this is possible by Archimede's axiom). Find
1780:, which represents the inversion of the line
1419:Line-circle intersection (non-collinear case)
147:Creating the line through two existing points
8:
2393:
1408:Circle center is not collinear with the line
723:if the circles intersect in only one point,
2377:
2375:
2373:
1501:(in red). (See above, compass equivalence.)
387:is its own reflection and lies on the line
2145:Motivated by Mascheroni's result, in 1822
1733:Under the assumption of this case, points
1632:then the line is tangential to the circle
1323:is inverted to the circle passing through
1266:is inverted to the circle passing through
2458:
2243:
2241:
2239:
1839:
1637:
1611:
1542:
850:
830:
802:
779:
759:
728:
670:
102:
98:
97:
94:
1867:Circle center is collinear with the line
2208:
1809:are the intersection points of circles
340:Construct two circles: one centered at
55:The result was originally published by
2587:Compass and straightedge constructions
2551:, Prometheus Books, pp. 197–216,
2276:Schogt, J. H. (1938) "Om Georg Mohr's
2105:Other types of restricted construction
2018:is the fourth vertex of parallelogram
1981:is the fourth vertex of parallelogram
1584:are the intersection points of circle
402:Extending the length of a line segment
2381:
2364:
2352:
1926:as the other intersection of circles
1754:of the circle passing through points
1000:is a positive integral multiple, say
31:can be performed by a compass alone.
7:
2422:
2337:
2328:, Journal of Geometry (1990) 38: 12.
2309:
2247:
2233:(Amsterdam: Jacob van Velsen, 1672).
2215:
1201:, find their point of intersection,
901:The light blue circles intersect at
67:until Mohr's work was rediscovered.
1475:Under the assumption of this case,
1460:, which is the reflection of point
992:so that the length of line segment
586:so that the length of line segment
2572:Construction with the Compass Only
2164:A proof later provided in 1904 by
2141:Restrictions excluding the compass
2110:Restrictions involving the compass
1860:, and the line is also tangential.
1566:Internal tangency is not possible.
224:and radius specified by a number,
139:To prove the theorem, each of the
14:
2506:Geometry / A Comprehensive Course
2486:The American Mathematical Monthly
2477:A Survey of Geometry (Volume One)
2263:, udkommet i Amsterdam i 1672" ,
2074:are the intersections of circles
1221:of arbitrary radius whose center
1087:Given three non-collinear points
774:simply by doubling the length of
335:upon reflection across this line.
122:(which cannot be formulated in a
2297:(Pavia: Pietro Galeazzi, 1797).
439:is the midpoint of line segment
306:Reflecting a point across a line
189:#3, #4 - The other constructions
181:#5 - Intersection of two circles
111:{\displaystyle \mathbb {R} ^{2}}
1992:as the intersection of circles
1955:as the intersection of circles
1698:, between them (if they exist).
1451:, between them (if they exist).
1352:be the intersection of circles
534:as the intersection of circles
495:as the intersection of circles
456:as the intersection of circles
78:An algebraic approach uses the
2498:10.1080/00029890.1994.11997027
2029:as an intersection of circles
1648:
1642:
278:Some preliminary constructions
205:and that passes through point
164:#1 - A line through two points
1:
2531:Circles / A Mathematical View
1103:of the circle they determine.
1042:is now obtained by extending
1504:The intersections of circle
1225:does not lie on either line.
521:is an equilateral triangle.)
482:is an equilateral triangle.)
270:geometric definition of the
63:in 1797 and it was known as
2134:compass equivalency theorem
797:(quadrupling the length of
754:, it is possible to invert
626:Point inversion in a circle
293:Compass equivalence theorem
2608:
2592:Theorems in plane geometry
924:is the desired inverse of
876:Construct two new circles
825:Reflect the circle center
290:
2295:La Geometria del Compasso
2127:Niccolò Fontana Tartaglia
1185:Given non-parallel lines
608:for any positive integer
600:⋅ length of line segment
524:Finally, construct point
2155:Poncelet–Steiner theorem
936:is such that the radius
73:inversion transformation
29:compass and straightedge
1515:and the new red circle
661:that is the inverse of
646:(in black) and a point
348:, both passing through
283:is listed prior to it.
220:. A circle with center
25:Mohr–Mascheroni theorem
1876:
1854:
1655:
1626:
1557:
1435:(in black) and a line
1420:
1181:
1083:
867:
839:
814:
791:
768:
748:
685:
627:
411:
315:
112:
2475:Eves, Howard (1963),
2459:Retz & Keihn 1989
2282:Matematisk Tidsskrift
2265:Matematisk Tidsskrift
1874:
1855:
1656:
1627:
1558:
1418:
1179:
1081:
959:is to the radius; or
905:and at another point
868:
840:
815:
792:
769:
749:
686:
625:
618:Inversion in a circle
415:Given a line segment
409:
319:Given a line segment
313:
113:
88:real coordinate space
2504:Pedoe, Dan (1988) ,
2293:Lorenzo Mascheroni,
2161:eleven years later.
2147:Jean Victor Poncelet
1838:
1654:{\displaystyle C(r)}
1636:
1610:
1541:
1456:Construct the point
1012:and is greater than
928:in the black circle.
849:
829:
801:
778:
758:
747:{\displaystyle E=E'}
727:
669:
657:construct the point
344:and one centered at
272:constructible number
228:, or a line segment
197:Notation and remarks
124:first-order language
93:
65:Mascheroni's Theorem
2197:Projective geometry
2151:projective geometry
1853:{\displaystyle P=Q}
1625:{\displaystyle P=Q}
1556:{\displaystyle P=Q}
1490:Construct a circle
1004:, of the length of
866:{\displaystyle EE'}
236:will be denoted by
209:will be denoted by
2192:Inversive geometry
2182:Napoleon's problem
1903:, find the points
1877:
1850:
1745:are not collinear.
1651:
1622:
1553:
1421:
1380:is the inverse of
1335:. Find the center
1278:. Find the center
1182:
1152:is the inverse of
1099:, find the center
1084:
863:
835:
813:{\displaystyle DB}
810:
790:{\displaystyle EB}
787:
764:
744:
681:
642:, for some radius
628:
412:
316:
130:Constructive proof
108:
61:Lorenzo Mascheroni
2558:978-1-63388-167-9
2540:978-0-88385-518-8
2515:978-0-486-65812-4
2479:, Allyn and Bacon
2394:Hungerbühler 1994
1895:lies on the line
1880:Given the circle
1769:Construct circle
1112:, the inverse of
838:{\displaystyle B}
767:{\displaystyle D}
120:Archimedes' axiom
2599:
2561:
2543:
2518:
2500:
2480:
2462:
2456:
2450:
2449:
2432:
2426:
2420:
2414:
2412:
2408:
2403:
2397:
2391:
2385:
2379:
2368:
2362:
2356:
2350:
2341:
2335:
2329:
2319:
2313:
2307:
2301:
2291:
2285:
2278:Euclides Danicus
2274:
2268:
2261:Euclides Danicus
2257:
2251:
2245:
2234:
2230:Euclides Danicus
2225:
2219:
2213:
2166:Francesco Severi
2157:, was proved by
2123:Gerolamo Cardano
2119:Lodovico Ferrari
2095:
2084:
2073:
2069:
2062:
2061:
2054:
2050:
2039:
2028:
2025:Construct point
2021:
2017:
2013:
2002:
1991:
1988:Construct point
1984:
1980:
1976:
1965:
1954:
1951:Construct point
1947:
1936:
1925:
1916:Construct point
1910:
1906:
1902:
1901:
1894:
1890:
1859:
1857:
1856:
1851:
1830:
1819:
1808:
1804:
1798:
1787:
1786:
1779:
1765:
1761:
1757:
1753:
1750:Find the center
1744:
1740:
1736:
1729:
1725:
1721:
1710:
1706:
1697:
1693:
1689:
1688:
1681:
1660:
1658:
1657:
1652:
1631:
1629:
1628:
1623:
1602:
1601:
1594:
1583:
1579:
1562:
1560:
1559:
1554:
1533:
1529:
1525:
1514:
1500:
1484:
1471:
1470:
1463:
1459:
1450:
1446:
1442:
1441:
1434:
1394:
1383:
1379:
1373:
1362:
1351:
1338:
1334:
1330:
1326:
1322:
1321:
1311:
1307:
1303:
1292:
1288:
1281:
1277:
1273:
1269:
1265:
1264:
1254:
1250:
1246:
1235:
1231:
1224:
1220:
1204:
1200:
1199:
1192:
1191:
1166:
1155:
1151:
1145:
1141:
1140:
1133:
1126:
1115:
1111:
1108:Construct point
1102:
1098:
1094:
1090:
1069:
1068:
1057:
1056:
1049:
1048:
1041:
1037:
1026:
1022:
1018:
1011:
1010:
1003:
999:
998:
991:
990:
983:
976:
958:
954:
950:
939:
935:
927:
923:
914:
904:
898:(in light blue).
897:
886:
872:
870:
869:
864:
862:
845:across the line
844:
842:
841:
836:
819:
817:
816:
811:
796:
794:
793:
788:
773:
771:
770:
765:
753:
751:
750:
745:
743:
717:
713:
706:
690:
688:
687:
684:{\textstyle D=B}
682:
664:
660:
656:
645:
641:
613:
607:
606:
599:
593:
592:
585:
577:
563:
555:
544:
533:
516:
505:
494:
485:Construct point
477:
466:
455:
452:Construct point
446:
445:
438:
434:
433:
426:
422:
421:
394:
393:
386:
382:
369:
368:
362:across the line
361:
357:
351:
347:
343:
334:
330:
326:
325:
264:circle inversion
258:, respectively.
257:
246:
235:
234:
227:
223:
219:
208:
204:
117:
115:
114:
109:
107:
106:
101:
2607:
2606:
2602:
2601:
2600:
2598:
2597:
2596:
2577:
2576:
2568:
2559:
2546:
2541:
2528:
2525:
2523:Further reading
2516:
2503:
2483:
2474:
2471:
2466:
2465:
2457:
2453:
2447:
2434:
2433:
2429:
2421:
2417:
2410:
2406:
2404:
2400:
2392:
2388:
2380:
2371:
2363:
2359:
2351:
2344:
2336:
2332:
2320:
2316:
2308:
2304:
2292:
2288:
2284:A, pages 34–36.
2275:
2271:
2258:
2254:
2246:
2237:
2226:
2222:
2214:
2210:
2205:
2178:
2143:
2117:mathematicians
2112:
2107:
2086:
2075:
2071:
2067:
2057:
2056:
2052:
2041:
2030:
2026:
2019:
2015:
2004:
1993:
1989:
1982:
1978:
1967:
1956:
1952:
1938:
1927:
1917:
1908:
1904:
1897:
1896:
1892:
1881:
1869:
1836:
1835:
1821:
1810:
1806:
1802:
1789:
1782:
1781:
1770:
1763:
1759:
1755:
1751:
1742:
1738:
1734:
1727:
1723:
1712:
1708:
1704:
1695:
1691:
1684:
1683:
1672:
1671:Given a circle
1634:
1633:
1608:
1607:
1597:
1596:
1585:
1581:
1577:
1539:
1538:
1531:
1527:
1516:
1505:
1491:
1476:
1472:. (See above.)
1466:
1465:
1461:
1457:
1448:
1444:
1437:
1436:
1425:
1424:Given a circle
1410:
1402:
1385:
1381:
1377:
1364:
1353:
1343:
1339:of this circle.
1336:
1332:
1328:
1324:
1317:
1316:
1309:
1305:
1294:
1290:
1286:
1282:of this circle.
1279:
1275:
1271:
1267:
1260:
1259:
1252:
1248:
1237:
1233:
1229:
1222:
1211:
1202:
1195:
1194:
1187:
1186:
1174:
1157:
1153:
1149:
1143:
1136:
1135:
1131:
1117:
1113:
1109:
1100:
1096:
1092:
1088:
1076:
1064:
1059:
1052:
1051:
1044:
1043:
1039:
1028:
1024:
1023:the inverse of
1020:
1013:
1006:
1005:
1001:
994:
993:
986:
985:
981:
960:
956:
952:
941:
937:
933:
925:
921:
906:
902:
888:
877:
855:
847:
846:
827:
826:
799:
798:
776:
775:
756:
755:
736:
725:
724:
715:
711:
697:
667:
666:
662:
658:
647:
643:
632:
631:Given a circle
620:
609:
602:
601:
595:
588:
587:
583:
578:are collinear.)
565:
561:
546:
535:
525:
507:
496:
486:
468:
457:
453:
441:
440:
436:
429:
428:
424:
417:
416:
404:
389:
388:
384:
374:
364:
363:
359:
355:
349:
345:
341:
332:
328:
321:
320:
308:
295:
289:
280:
248:
237:
230:
229:
225:
221:
210:
206:
202:
199:
137:
132:
96:
91:
90:
84:Euclidean plane
53:
17:
12:
11:
5:
2605:
2603:
2595:
2594:
2589:
2579:
2578:
2575:
2574:
2567:
2566:External links
2564:
2563:
2562:
2557:
2544:
2539:
2524:
2521:
2520:
2519:
2514:
2501:
2492:(8): 784–787,
2481:
2470:
2467:
2464:
2463:
2451:
2445:
2427:
2415:
2398:
2386:
2369:
2357:
2342:
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2302:
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2269:
2252:
2235:
2220:
2207:
2206:
2204:
2201:
2200:
2199:
2194:
2189:
2187:Geometrography
2184:
2177:
2174:
2142:
2139:
2111:
2108:
2106:
2103:
2098:
2097:
2064:
2023:
1986:
1949:
1913:
1912:
1868:
1865:
1864:
1863:
1862:
1861:
1849:
1846:
1843:
1800:
1767:
1748:
1747:
1746:
1730:respectively.
1703:Invert points
1700:
1699:
1665:
1664:
1663:
1662:
1650:
1647:
1644:
1641:
1621:
1618:
1615:
1574:
1573:
1572:
1569:
1568:
1567:
1552:
1549:
1546:
1502:
1488:
1487:
1486:
1453:
1452:
1409:
1406:
1401:
1398:
1397:
1396:
1384:in the circle
1375:
1340:
1313:
1285:Invert points
1283:
1256:
1228:Invert points
1226:
1210:Select circle
1207:
1206:
1173:
1170:
1169:
1168:
1156:in the circle
1147:
1128:
1116:in the circle
1105:
1104:
1075:
1072:
930:
929:
918:
917:
916:
899:
861:
858:
854:
834:
823:
822:
821:
809:
806:
786:
783:
763:
742:
739:
735:
732:
721:
708:
696:Draw a circle
693:
692:
680:
677:
674:
619:
616:
580:
579:
522:
483:
449:
448:
403:
400:
399:
398:
397:
396:
353:
337:
336:
314:Point symmetry
307:
304:
291:Main article:
288:
285:
279:
276:
198:
195:
161:
160:
157:
154:
151:
148:
136:
133:
131:
128:
105:
100:
52:
49:
44:
43:
15:
13:
10:
9:
6:
4:
3:
2:
2604:
2593:
2590:
2588:
2585:
2584:
2582:
2573:
2570:
2569:
2565:
2560:
2554:
2550:
2545:
2542:
2536:
2532:
2527:
2526:
2522:
2517:
2511:
2507:
2502:
2499:
2495:
2491:
2487:
2482:
2478:
2473:
2472:
2468:
2460:
2455:
2452:
2448:
2446:9780873532815
2442:
2438:
2431:
2428:
2424:
2419:
2416:
2402:
2399:
2395:
2390:
2387:
2383:
2378:
2376:
2374:
2370:
2366:
2361:
2358:
2354:
2349:
2347:
2343:
2339:
2334:
2331:
2327:
2323:
2318:
2315:
2311:
2306:
2303:
2300:
2299:1901 edition.
2296:
2290:
2287:
2283:
2279:
2273:
2270:
2267:B, pages 1–7.
2266:
2262:
2256:
2253:
2249:
2244:
2242:
2240:
2236:
2232:
2231:
2224:
2221:
2217:
2212:
2209:
2202:
2198:
2195:
2193:
2190:
2188:
2185:
2183:
2180:
2179:
2175:
2173:
2169:
2167:
2162:
2160:
2159:Jakob Steiner
2156:
2152:
2148:
2140:
2138:
2135:
2130:
2128:
2124:
2120:
2116:
2109:
2104:
2102:
2093:
2089:
2082:
2078:
2065:
2060:
2048:
2044:
2037:
2033:
2024:
2011:
2007:
2000:
1996:
1987:
1974:
1970:
1963:
1959:
1950:
1945:
1941:
1934:
1930:
1924:
1920:
1915:
1914:
1900:
1891:whose center
1888:
1884:
1879:
1878:
1873:
1866:
1847:
1844:
1841:
1833:
1832:
1828:
1824:
1817:
1813:
1801:
1796:
1792:
1785:
1777:
1773:
1768:
1749:
1732:
1731:
1719:
1715:
1702:
1701:
1687:
1679:
1675:
1670:
1669:
1668:
1645:
1639:
1619:
1616:
1613:
1605:
1604:
1600:
1595:and the line
1592:
1588:
1575:
1570:
1565:
1564:
1550:
1547:
1544:
1536:
1535:
1523:
1519:
1512:
1508:
1503:
1498:
1494:
1489:
1483:
1479:
1474:
1473:
1469:
1455:
1454:
1440:
1432:
1428:
1423:
1422:
1417:
1413:
1407:
1405:
1399:
1392:
1388:
1376:
1371:
1367:
1360:
1356:
1350:
1346:
1341:
1320:
1314:
1312:respectively.
1301:
1297:
1284:
1263:
1257:
1255:respectively.
1244:
1240:
1227:
1218:
1214:
1209:
1208:
1198:
1190:
1184:
1183:
1178:
1171:
1164:
1160:
1148:
1142:to the point
1139:
1129:
1124:
1120:
1107:
1106:
1086:
1085:
1080:
1073:
1071:
1067:
1062:
1055:
1047:
1035:
1031:
1016:
1009:
997:
989:
978:
975:
971:
967:
963:
948:
944:
919:
913:
909:
900:
895:
891:
884:
880:
875:
874:
859:
856:
852:
832:
824:
807:
804:
784:
781:
761:
740:
737:
733:
730:
722:
719:
718:
709:
704:
700:
695:
694:
678:
675:
672:
654:
650:
639:
635:
630:
629:
624:
617:
615:
612:
605:
598:
591:
576:
572:
568:
559:
553:
549:
542:
538:
532:
528:
523:
520:
514:
510:
503:
499:
493:
489:
484:
481:
475:
471:
464:
460:
451:
450:
444:
432:
423:find a point
420:
414:
413:
408:
401:
392:
381:
377:
372:
371:
367:
354:
339:
338:
324:
318:
317:
312:
305:
303:
301:
294:
286:
284:
277:
275:
273:
267:
265:
259:
255:
251:
244:
240:
233:
217:
213:
196:
194:
191:
190:
186:
183:
182:
178:
175:
174:
170:
166:
165:
158:
155:
152:
149:
146:
145:
144:
142:
134:
129:
127:
125:
121:
103:
89:
85:
81:
76:
74:
68:
66:
62:
58:
50:
48:
42:
38:
37:
36:
32:
30:
26:
22:
2548:
2530:
2505:
2489:
2485:
2476:
2454:
2436:
2430:
2418:
2401:
2389:
2360:
2333:
2317:
2305:
2294:
2289:
2281:
2277:
2272:
2264:
2260:
2255:
2228:
2227:Georg Mohr,
2223:
2211:
2170:
2163:
2144:
2131:
2113:
2099:
2091:
2087:
2080:
2076:
2058:
2046:
2042:
2035:
2031:
2009:
2005:
1998:
1994:
1972:
1968:
1961:
1957:
1943:
1939:
1932:
1928:
1922:
1918:
1898:
1886:
1882:
1826:
1822:
1815:
1811:
1794:
1790:
1788:into circle
1783:
1775:
1771:
1717:
1713:
1685:
1677:
1673:
1666:
1598:
1590:
1586:
1521:
1517:
1510:
1506:
1496:
1492:
1481:
1477:
1467:
1464:across line
1438:
1430:
1426:
1411:
1403:
1390:
1386:
1369:
1365:
1358:
1354:
1348:
1344:
1318:
1299:
1295:
1261:
1242:
1238:
1216:
1212:
1196:
1188:
1162:
1158:
1137:
1134:in the line
1122:
1118:
1065:
1060:
1053:
1045:
1033:
1029:
1014:
1007:
995:
987:
984:on the line
979:
973:
969:
965:
961:
946:
942:
931:
911:
907:
893:
889:
882:
878:
702:
698:
652:
648:
637:
633:
610:
603:
596:
589:
581:
574:
570:
566:
557:
551:
547:
540:
536:
530:
526:
518:
512:
508:
501:
497:
491:
487:
479:
473:
469:
462:
458:
442:
430:
427:on the line
418:
390:
379:
375:
365:
327:and a point
322:
300:The Elements
299:
296:
281:
268:
260:
253:
249:
242:
238:
231:
215:
211:
200:
192:
188:
187:
184:
180:
179:
176:
172:
171:
167:
163:
162:
138:
82:between the
77:
69:
64:
54:
45:
39:
33:
24:
18:
2322:Arnon Avron
2115:Renaissance
1682:and a line
1526:are points
80:isomorphism
21:mathematics
2581:Categories
2549:The Circle
2469:References
2382:Pedoe 1988
2365:Pedoe 1988
2353:Pedoe 1988
1722:to points
1711:in circle
1304:to points
1293:in circle
1247:to points
1236:in circle
1027:in circle
564:show that
435:such that
57:Georg Mohr
2508:, Dover,
2423:Eves 1963
2338:Eves 1963
2310:Eves 1963
2248:Eves 1963
2216:Eves 1963
1315:The line
1258:The line
707:(in red).
41:together.
2461:, p. 196
2425:, p. 200
2396:, p. 784
2384:, p. 123
2340:, p. 184
2312:, p. 198
2250:, p. 199
2218:, p. 201
2176:See also
2055:lies on
1130:Reflect
1050:so that
860:′
741:′
86:and the
2367:, p. 77
2355:, p. 78
2066:Points
1576:Points
135:Outline
51:History
2555:
2537:
2512:
2443:
2020:CDD'F'
1762:, and
1741:, and
951:is to
932:Point
920:Point
23:, the
2203:Notes
1983:CD'DF
2553:ISBN
2535:ISBN
2510:ISBN
2441:ISBN
2409:and
2132:The
2125:and
2085:and
2070:and
2040:and
2003:and
1999:DD'
1966:and
1962:DD'
1937:and
1907:and
1820:and
1805:and
1726:and
1707:and
1694:and
1580:and
1530:and
1447:and
1363:and
1342:Let
1331:and
1308:and
1289:and
1274:and
1251:and
1232:and
1193:and
1095:and
1066:BQ'
1046:BQ'
887:and
714:and
573:and
556:. (∆
545:and
517:. (∆
506:and
478:. (∆
467:and
2494:doi
2490:101
2280:,"
2051:. (
2043:F'
2036:D'
2014:. (
2006:D'
1977:. (
1919:D'
1606:If
1017:/ 2
955:as
940:of
890:E'
651:(≠
558:EBC
519:DBE
480:ABD
373:If
247:or
126:).
19:In
2583::
2488:,
2372:^
2345:^
2324:,
2238:^
2121:,
2081:CM
2063:.)
2059:AB
2022:.)
2016:F'
1990:F'
1985:.)
1921:≠
1899:AB
1831:.
1784:AB
1764:B'
1760:A'
1758:,
1739:B'
1737:,
1735:A'
1728:B'
1724:A'
1686:AB
1603:.
1599:AB
1563:.
1534:.
1480:≠
1468:AB
1439:AB
1347:≠
1333:D'
1329:C'
1327:,
1319:CD
1310:D'
1306:C'
1276:B'
1272:A'
1270:,
1262:AB
1253:B'
1249:A'
1197:CD
1189:AB
1138:BD
1091:,
1070:.
1063:⋅
1058:=
1054:BI
1021:Q'
1008:BD
996:BQ
988:BD
977:.
974:DB
972:/
968:=
964:/
962:IB
957:DB
953:IB
910:≠
873::
820:).
716:E'
614:.
604:AB
594:=
590:AQ
569:,
529:≠
490:≠
443:AC
431:AB
419:AB
391:AB
378:=
370:.
366:AB
323:AB
254:AB
232:AB
75:.
2496::
2411:Q
2407:P
2096:.
2094:)
2092:D
2090:(
2088:C
2083:)
2079:(
2077:F
2072:Q
2068:P
2053:M
2049:)
2047:D
2045:(
2038:)
2034:(
2032:F
2027:M
2012:)
2010:C
2008:(
2001:)
1997:(
1995:C
1979:F
1975:)
1973:C
1971:(
1969:D
1964:)
1960:(
1958:C
1953:F
1948:.
1946:)
1944:D
1942:(
1940:C
1935:)
1933:D
1931:(
1929:A
1923:D
1909:Q
1905:P
1893:C
1889:)
1887:D
1885:(
1883:C
1848:Q
1845:=
1842:P
1829:)
1827:C
1825:(
1823:E
1818:)
1816:r
1814:(
1812:C
1807:Q
1803:P
1799:.
1797:)
1795:r
1793:(
1791:C
1778:)
1776:C
1774:(
1772:E
1766:.
1756:C
1752:E
1743:C
1720:)
1718:r
1716:(
1714:C
1709:B
1705:A
1696:Q
1692:P
1680:)
1678:r
1676:(
1674:C
1661:.
1649:)
1646:r
1643:(
1640:C
1620:Q
1617:=
1614:P
1593:)
1591:r
1589:(
1587:C
1582:Q
1578:P
1551:Q
1548:=
1545:P
1532:Q
1528:P
1524:)
1522:r
1520:(
1518:D
1513:)
1511:r
1509:(
1507:C
1499:)
1497:r
1495:(
1493:D
1485:.
1482:D
1478:C
1462:C
1458:D
1449:Q
1445:P
1433:)
1431:r
1429:(
1427:C
1395:.
1393:)
1391:r
1389:(
1387:O
1382:Y
1378:X
1374:.
1372:)
1370:O
1368:(
1366:F
1361:)
1359:O
1357:(
1355:E
1349:O
1345:Y
1337:F
1325:O
1302:)
1300:r
1298:(
1296:O
1291:D
1287:C
1280:E
1268:O
1245:)
1243:r
1241:(
1239:O
1234:B
1230:A
1223:O
1219:)
1217:r
1215:(
1213:O
1205:.
1203:X
1167:.
1165:)
1163:B
1161:(
1159:A
1154:X
1150:O
1146:.
1144:X
1132:A
1127:.
1125:)
1123:B
1121:(
1119:A
1114:C
1110:D
1101:O
1097:C
1093:B
1089:A
1061:n
1040:I
1036:)
1034:r
1032:(
1030:B
1025:Q
1015:r
1002:n
982:Q
970:r
966:r
949:)
947:r
945:(
943:B
938:r
934:I
926:D
922:I
915:.
912:B
908:I
903:B
896:)
894:B
892:(
885:)
883:B
881:(
879:E
857:E
853:E
833:B
808:B
805:D
785:B
782:E
762:D
738:E
734:=
731:E
712:E
705:)
703:B
701:(
699:D
691:.
679:B
676:=
673:D
663:D
659:I
655:)
653:B
649:D
644:r
640:)
638:r
636:(
634:B
611:n
597:n
584:Q
575:C
571:B
567:A
562:B
554:)
552:B
550:(
548:E
543:)
541:E
539:(
537:B
531:D
527:C
515:)
513:D
511:(
509:B
504:)
502:B
500:(
498:D
492:A
488:E
476:)
474:A
472:(
470:B
465:)
463:B
461:(
459:A
454:D
447:.
437:B
425:C
385:C
380:D
376:C
360:C
356:D
352:.
350:C
346:B
342:A
333:C
329:C
256:)
252:(
250:U
245:)
243:r
241:(
239:U
226:r
222:U
218:)
216:V
214:(
212:U
207:V
203:U
104:2
99:R
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