X19

CLAWSON POINT

X19 is the homothetic center of the orthic and extangents triangles. The Ayme triangle, constructed at X3610, is perspective to ABC with perspector X19.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 4, 9 }: Orthocenter–Spieker Line { 1, 28 }: Central Line L(1, 28) { 6, 34 }: Central Line L(6, 34) { 25, 33 }: Central Line L(25, 33) { 27, 63 }: Central Line L(27, 63) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_4 = ORTHOCENTER (0.6122, 0.3958) X_9 = MITTENPUNKT (0.5110, 0.1615) X_1 = INCENTER (0.5714, 0.2333) X_28 = CEVAPOINT OF X(19) AND X(25) (0.6285, 0.4382) X_6 = SYMMEDIAN POINT (LEMOINE POINT, GREBE POINT) (0.6000, 0.2672) X_34 = -BETH CONJUGATE OF X(4) (0.6332, 0.4792) X_25 = HOMOTHETIC CENTER OF ORTHIC AND TANGENTIAL TRIANGLES (0.6329, 0.4497) X_33 = PERSPECTOR OF THE ORTHIC AND INTANGENTS TRIANGLES (0.6019, 0.3546) X_27 = CEVAPOINT OF ORTHOCENTER AND CLAWSON CENTER (0.6182, 0.4112) X_63 = ISOGONAL CONJUGATE OF X(19) (0.4566, 0.0823) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_19 = CLAWSON POINT (0.6246, 0.4244) A B C X4 X9 X1 X28 X6 X34 X25 X33 X27 X63 X10 X19

X19 (CLAWSON POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\tan A : \tan B : \tan C\)
Barycentrics: \(a (a^{2} + b^{2} - c^{2}) \cdot (a^{2} - b^{2} + c^{2}) : :\)
Tripolars: \((-a^{2}+b^{2}+c^{2}) \sqrt{b c (a^{6}-a^{4} b^{2}-a^{2} b^{4}+b^{6}-2 a^{4} b c-2 b^{5} c-a^{4} c^{2}+2 a^{2} b^{2} c^{2}-b^{4} c^{2}+4 b^{3} c^{3}-a^{2} c^{4}-b^{2} c^{4}-2 b c^{5}+c^{6})} : :\)
Search Value (6, 9, 13): \(-1.480512397789\)
▸ Alternate Coordinates5 forms

  • Trilinears: \(\sin 2B + \sin 2C - \sin 2A : :\)

  • Trilinears: \(1/(b^{2} + c^{2} - a^{2}) : :\)

  • Trilinears: \(a^{2} - S^{2} + S_{A} : :\)

  • Trilinears: \(S_{B} S_{C} : :\)

  • Barycentrics: \(a \tan A : b \tan B : c \tan C\)

▸ Linear Combinations2 combinations

  • X19 = \((r + 2R - s)(r + 2R + s)\) X1 - \(6R(r + 2R)\) X2 - \(2(r^{2} + 2rR - s^{2})\) X3 (Peter Moses, April 2, 2013)

  • X19 = \(3\) X2 + X20061, \(6\) X2 - \(5\) X31261, \(2\) X3 - \(3\) X21160, \(2\) X4329 - \(3\) X31158, \(2\) X4329 - \(5\) X31261, \(2\) X18589 + X20061, \(4\) X18589 - \(3\) X31158, \(4\) X18589 - \(5\) X31261, \(2\) X20061 + \(3\) X31158, \(2\) X20061 + \(5\) X31261, \(3\) X21160 - X30265, \(3\) X31158 - \(5\) X31261

X19 lies on the 2nd Lester circle (Yiu), the cubics K109, K145, K175, K343, K391, K431, K445, K457, K605, K678, K696, K697, K750, K864, K968, K999, K1039, K1042, K1090, the curve Q121.

Related Collection 1 Topic Guides

T_12: The Fuhrmann & Spieker Central Lines • T_19: The Brocard & Orthocentroidal Circles • T_21: Orthic & Tangential Triangles

▸ Incident Central Lines395 lines

{1, 28}, {2, 534}, {3, 1871}, {4, 9}, {5, 8141}, {6, 34}, {7, 5236}, {8, 1891}, {21, 4288}, {24, 2337}, {25, 33}, {27, 63}, {29, 5250}, {30, 15940}, {31, 204}, {32, 5336}, {35, 14017}, {38, 4211}, {41, 1825}, {44, 1828}, {45, 1900}, {46, 579}, {47, 921}, {51, 3611}, {52, 6237}, {53, 1846}, {54, 16031}, {56, 207}, {57, 196}, {64, 1903}, {77, 2002}, {81, 969}, {84, 3183}, {86, 14013}, {91, 920}, {100, 7466}, {101, 913}, {102, 282}, {107, 2249}, {108, 2291}, {109, 8775}, {112, 759}, {113, 12661}, {117, 5190}, {125, 10119}, {143, 32158}, {155, 12417}, {158, 1712}, {162, 897}, {163, 563}, {165, 2954}, {184, 10536}, {185, 6254}, {186, 16553}, {191, 31902}, {200, 3949}, {208, 225}, {216, 27622}, {219, 517}, {220, 1902}, {226, 1763}, {232, 444}, {264, 21371}, {269, 3942}, {270, 2363}, {273, 653}, {275, 19181}, {294, 1041}, {297, 25978}, {307, 24316}, {318, 1840}, {326, 1958}, {331, 10030}, {346, 3610}, {347, 24604}, {355, 7511}, {378, 7688}, {379, 1441}, {381, 18453}, {403, 7110}, {406, 5257}, {407, 1865}, {418, 26908}, {423, 2905}, {427, 3925}, {428, 5101}, {429, 1213}, {469, 3305}, {475, 5750}, {484, 5146}, {487, 12662}, {488, 12663}, {511, 15975}, {518, 5781}, {523, 14119}, {560, 1910}, {577, 1950}, {583, 2969}, {587, 3536}, {594, 5090}, {596, 14964}, {604, 909}, {614, 3162}, {648, 18827}, {649, 3064}, {650, 2432}, {656, 8768}, {662, 8771}, {672, 1851}, {774, 2155}, {775, 4100}, {798, 24006}, {800, 1945}, {823, 1821}, {846, 17038}, {851, 18591}, {876, 2489}, {908, 28807}, {911, 17463}, {960, 965}, {962, 27382}, {990, 3220}, {993, 4227}, {1013, 35258}, {1024, 7649}, {1030, 20832}, {1039, 2298}, {1040, 4224}, {1075, 7554}, {1086, 28017}, {1100, 11396}, {1104, 3172}, {1109, 2157}, {1122, 2097}, {1125, 7521}, {1158, 1715}, {1174, 2266}, {1212, 1593}, {1214, 11347}, {1247, 2959}, {1319, 37519}, {1333, 2217}, {1334, 28076}, {1350, 5784}, {1375, 17073}, {1405, 1866}, {1422, 7099}, {1423, 3512}, {1449, 1870}, {1457, 22063}, {1460, 7337}, {1461, 34492}, {1482, 20818}, {1572, 2300}, {1580, 17891}, {1581, 1740}, {1598, 1872}, {1604, 11399}, {1609, 2164}, {1611, 3290}, {1621, 4233}, {1633, 1721}, {1659, 3068}, {1698, 5142}, {1699, 9572}, {1707, 1719}, {1708, 1713}, {1709, 8558}, {1743, 1783}, {1745, 23619}, {1759, 21061}, {1773, 13161}, {1802, 2324}, {1830, 2246}, {1831, 2268}, {1836, 1901}, {1837, 1852}, {1843, 2876}, {1847, 10509}, {1857, 2357}, {1862, 6154}, {1864, 17810}, {1877, 2347}, {1878, 5183}, {1883, 17369}, {1944, 10446}, {1964, 2186}, {1966, 18832}, {1968, 16968}, {1969, 3403}, {1974, 16972}, {1986, 7724}, {1990, 2160}, {2128, 2129}, {2148, 2190}, {2156, 17871}, {2159, 3708}, {2166, 3376}, {2192, 7007}, {2194, 2219}, {2195, 2212}, {2203, 2214}, {2204, 2218}, {2207, 2281}, {2215, 8747}, {2255, 7151}, {2256, 3057}, {2258, 3192}, {2267, 30503}, {2273, 9620}, {2278, 17443}, {2287, 3869}, {2290, 2962}, {2305, 2652}, {2313, 9251}, {2319, 3186}, {2321, 17742}, {2326, 13739}, {2578, 2588}, {2579, 2589}, {2809, 8271}, {2822, 5667}, {2911, 21853}, {2960, 4877}, {3059, 7716}, {3060, 11445}, {3062, 10859}, {3069, 6203}, {3091, 9537}, {3144, 34920}, {3169, 34895}, {3174, 16550}, {3176, 10396}, {3189, 7718}, {3199, 21796}, {3207, 11363}, {3211, 24474}, {3219, 6994}, {3247, 4262}, {3306, 16706}, {3330, 7355}, {3333, 22088}, {3400, 3408}, {3401, 3409}, {3402, 3404}, {3462, 7344}, {3497, 3500}, {3567, 11460}, {3574, 32370}, {3575, 6253}, {3576, 7501}, {3589, 5834}, {3666, 15509}, {3683, 11323}, {3692, 5174}, {3694, 5687}, {3729, 20602}, {3731, 4222}, {3772, 16318}, {3811, 22021}, {3875, 31906}, {3877, 17519}, {3958, 12526}, {4063, 17924}, {4183, 4512}, {4194, 5296}, {4200, 5749}, {4209, 7131}, {4212, 17754}, {4228, 20243}, {4254, 11398}, {4295, 5746}, {4320, 17807}, {4327, 22769}, {4360, 31910}, {4361, 5792}, {4384, 16566}, {4394, 6591}, {4429, 5125}, {4466, 18634}, {5095, 32277}, {5130, 17275}, {5155, 17330}, {5200, 13427}, {5248, 25081}, {5282, 7102}, {5292, 34266}, {5322, 10829}, {5412, 5415}, {5413, 5416}, {5437, 17917}, {5521, 20623}, {5739, 8896}, {5742, 26066}, {5747, 12047}, {5776, 6001}, {5778, 5887}, {5802, 18391}, {5813, 28739}, {5829, 5880}, {5928, 13567}, {6059, 7083}, {6152, 6255}, {6180, 34371}, {6252, 6291}, {6404, 6406}, {6763, 31901}, {7017, 17787}, {7054, 11101}, {7190, 18162}, {7282, 8545}, {7350, 7351}, {7359, 12699}, {7412, 10268}, {7498, 31435}, {7534, 26921}, {7537, 8227}, {7559, 7989}, {7714, 34607}, {7982, 22356}, {7994, 14493}, {8148, 22147}, {8539, 8541}, {8743, 16470}, {8897, 18134}, {9310, 17452}, {9786, 12664}, {9817, 33849}, {10222, 23073}, {10311, 10315}, {10436, 15149}, {10476, 22065}, {10636, 10641}, {10637, 10642}, {10977, 33630}, {10985, 10988}, {11341, 20172}, {11433, 18921}, {12135, 17299}, {12329, 21867}, {13041, 13051}, {13042, 13052}, {14543, 17220}, {14557, 34048}, {14953, 17134}, {15344, 28847}, {16502, 20227}, {16551, 20367}, {16571, 17799}, {16572, 34498}, {16747, 32092}, {16814, 17516}, {17151, 31918}, {17465, 34080}, {17747, 28070}, {17861, 21364}, {17904, 20083}, {18163, 18677}, {18206, 31919}, {18344, 23351}, {18650, 24683}, {18679, 27659}, {18691, 20902}, {19213, 19214}, {19215, 19218}, {19216, 19217}, {19298, 19300}, {19299, 19301}, {19302, 21773}, {19432, 19446}, {19433, 19447}, {20258, 24334}, {20266, 21621}, {20291, 31015}, {20305, 24682}, {21368, 26665}, {21442, 24587}, {21770, 34434}, {22124, 34040}, {22127, 35631}, {22840, 22970}, {24315, 34830}, {24684, 25523}, {24701, 25365}, {25078, 25440}, {25590, 31925}, {26118, 34822}, {26273, 28023}, {26690, 35974}, {26704, 29068}, {26919, 26952}, {27472, 31346}, {31903, 32922}, {33790, 33793}, {36131, 36151}

▸ Functional Relations66 relations

Notes

Although John Clawson (../bstud/clawson.html) studied this point in 1925, it was studied earlier by Lemoine:

Emile Lemoine, “Quelques questions se rapportant à l’étude des antiparallèles des côtes d’un triangle”, Bulletin de la S. M. F., tome 14 (1886), p. 107-128, specifically, on page 114. This article is available online at Numdam.

Let A'B'C' be the 4th Brocard triangle. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X19. (Randy Hutson, November 18, 2015)

Let HaHbHc be the orthic triangle. Let A″ be the trilinear product of the (real or imaginary) circumcircle intercepts of line HbHc. Define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X19. (Randy Hutson, December 26, 2015)

Let JaJbJc be the excentral triangle. Let A″ be the pole, wrt the polar circle, of line JbJc, and define B″ and C″ cyclically. The lines JaA″, JbB″, JcC″ concur in X19. (Randy Hutson, December 2, 2017)

Let A'B'C' be the hexyl triangle. Let La be the trilinear polar of A', and define Lb and Lc cyclically. Let A″ = Lb∩Lc, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X19. (Randy Hutson, December 2, 2017)

Let A'B'C' be the hexyl triangle. Let Ab = BC∩C'A', Ac = BC∩A'B', and define Bc, Ba, Ca, Cb cyclically. Then Ab, Ac, Bc, Ba, Ca, Cb lie on an ellipse. Let A″ be the intersection of the tangents to the ellipse at Ba and Ca, and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X19. (Randy Hutson, December 2, 2017)

Let A'B'C' be the intouch triangle of the extangents triangle, if ABC is acute. Then A'B'C' is homothetic to the anti-Ara triangle at X19. (Randy Hutson, December 2, 2017)

Let La be the A-extraversion of line X650X663 (the trilinear polar of X9), and define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X19. (Randy Hutson, December 2, 2017)

Let La be the A-extraversion of line X661X663 (the trilinear polar of X19), and define Lb and Lc cyclically. Let A' = Lb∩Lc, and define B' and C' cyclically. The lines AA', BB', CC' concur in X19. (Randy Hutson, December 2, 2017)