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 , select the center name from the list and, then, click on the vertices A, B and C successively.
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 ŕ l'étude des antiparallčles 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 1837,1852X(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)