## X(40) (BEVAN POINT)

 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.

 The JRE (Java Runtime Environment) is not enabled in your browser!

This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           cos B + cos C - cos A - 1 : cos C + cos A - cos B - 1 : cos A + cos B - cos C - 1
= f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = b/(c + a - b) + c/(a + b - c) - a/(b + c - a)
= g(A,B,C) : g(B,C,A) : g(C,A,B), where
g(A,B,C) = sin2(B/2) + sin2(C/2) - sin2(A/2)

Barycentrics    af(a,b,c) : bf(b,c,a) : cf(c,a,b)

X(40) = point of concurrence of the perpendiculars from the excenters to the respective sides
X(40) = circumcenter of the excentral triangle
X(40) = incenter of the extangents triangle if triangle ABC is acute
X(40) = perspector of the excentral and extangents triangles

This point is mentioned in a problem proposal by Benjamin Bevan, published in Leybourn's Mathematical Repository, 1804, p. 18.

X(40) lies on these lines:
1,3    2,926    4,9    6,380    8,20    30,191    31,580    33,201    34,212    42,581    43,970    58,601    64,72    77,947    78,100    80,90    92,412    101,972    108,207    109,255    164,188    190,341    196,208    219,610    220,910    221,223    256,989    376,519    386,1064    387,579    390,938    392,474    511,1045    550,952    595,602    728,1018    936,960    958,1012    978,1050

X(40) is the {X(55),X(65)}-harmonic conjugate of X(1).

X(40) = midpoint of X(8) and X(20)
X(40) = reflection of X(I) in X(J) for these (I,J): (1,3), (4,10), (84,1158), (962,946), (1482,1385)
X(40) = isogonal conjugate of X(84)
X(40) = isotomic conjugate of X(309)
X(40) = complement of X(962)
X(40) = anticomplement of X(946)
X(40) = X(I)-Ceva conjugate of X(J) for these (I,J): (8,1), (20,1490), (63,9), (347,223)
X(40) = X(I)-cross conjugate of X(J) for these (I,J): (198,223), (221,1)
X(40) = crosspoint of X(I) and X(J) for these (I,J): (329,347)
X(40) = crosssum of X(56) and X(1413)
X(40) = crossdifference of any two points on line X(650)X(1459)
X(40) = X(I)-aleph conjugate of X(J) for these (I,J): (1,978), (2,57), (8,40), (188,1), (556,63)
X(40) = X(I)-beth conjugate of X(J) for these (I,J): (8,4), (40,221), (643,78), (644,728)

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