INSTITUTO DE MATEMÁTICA
HJB --- GMA --- UFF

X(19)
(CLAWSON POINT)


Click here to access the list of all triangle centers.

Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon Run Macro Tool, select the center name from the list and, then, click on the vertices A, B and C successively.

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Download all construction files and macros: tc.zip (10.1 Mb).
This applet was built with the free and multiplatform dynamic geometry software C.a.R..


Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           tan A : tan B : tan C
                                    = f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = sin 2B + sin 2C - sin 2A
                                    = g(a,b,c) : g(b,c,a) : g(c,a,b), where g(a,b,c) = 1/(b2 + c2 - a2)

Barycentrics    a tan A : b tan B : c tan C

X(19) is the homothetic center of the orthic and extangents triangles.

Further information is available from
Paul Yiu's Website.

Although John Clawson 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.

X(19) lies on these lines:
1,28    2,534    3,1871    4,9    6,34    8,1891    25,33    27,63    31,204    41,1825    44,1828    45,1900    46,579    47,921    53,1846    56,207    57,196    64,1903    81,969    91,920    101,913    102,282    112,759    158,1712    162,897    163,563    208,225    219,517    220,1902    226,1763    232,444    273,653    294,1041    318,1840    379,1441    407,1865    429,1213    560,1910    604,909    672,1851    960,965    1158,1715    1212,1593    1405,1866    1449,1870    1581,1740    1598,1872    1633,1721    1707,1719    1708,1713    1743,1783    1836,1901    1837,1852

X(19) is the {X(607),X(608)}-harmonic conjugate of X(6).

X(19) = isogonal conjugate of X(63)
X(19) = isotomic conjugate of X(304)

X(19) = X(I)-Ceva conjugate of X(J) for these (I,J):
(1,204), (4,33), (27,4), (28,25), (57,208), (92,1), (196,207), (278,34)

X(19) = X(I)-cross conjugate of X(J) for these (I,J): (25,34), (31,1)
X(19) = crosspoint of X(I) and X(J) for these (I,J): (4,278), (27,28), (57,84), (92,158)
X(19) = crosssum of X(I) and X(J) for these (I,J): (1,610), (3,219), (9,40), (48,255), (71,72)
X(19) = crossdifference of any two points on line X(521)X(656)
X(19) = X(I)-Hirst inverse of X(J) for these (I,J): (1,240), (4,242)
X(19) = X(I)-aleph conjugate of X(J) for these (I,J): (2,610), (92,19), (508,223), (648,163)
X(19) = X(I)-beth conjugate of X(J) for these (I,J): (9,198), (19,608), (112,604), (281,281), (648,273), (653,19)


This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.

If you have questions or suggestions, please, contact us using the e-mail presented here.

Departamento de Matemática Aplicada -- Instituto de Matemática -- Universidade Federal Fluminense




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