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)
, select the center name from the list and, then, click on the vertices A, B and C successively.