## X(4) (ORTHOCENTER)

 Interactive Applet

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 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           sec A : sec B : sec C
= cos A - sin B sin C : cos B - sin C sin A : cos C - sin A sinB
= cos A - cos(B - C) : cos B - cos(C - A) : cos C - cos(A - B)
= sin B sin C - cos(B - C) : sin C sin A - cos(C - A) : sin A sin B - cos(A - B)

Barycentrics    tan A : tan B : tan C

X(4) is the point of concurrence of the altitudes of ABC.

X(4) and the vertices A,B,C comprise an orthocentric system, defined as a set of four points, one of which is the orthocenter of the triangle of the other three (so that each is the orthocenter of the other three). The incenter and excenters also form an orthocentric system. If ABC is acute, then X(4) is the incenter of the orthic triangle.

Suppose P is not on a sideline of ABC. Let A'B'C' be the cevian triangle of P. Let D,E,F be the circles having diameters AA', BB',CC'. The radical center of D,E,F is X(4). (Floor van Lamoen, Hyacinthos #214, Jan. 24, 2000.)

Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 2: The Orthocenter.

X(4) is the {X(3),X(5)}-harmonic conjugate of X(2).

X(4) = midpoint of X(I) and X(J) for these (I,J):
(3,382), (147,148), (149,153), (150,152)

X(4) = reflection of X(I) in X(J) for these (I,J): (1,946), (2,381), (3,5), (5,546), (8,355), (20,3), (24,235), (25,1596), (40,10), (69,1352), (74,125), (98,115), (99,114), (100,119), (101,118), (102,124), (103,116), (104,11), (107,133), (109,117), (110,113), (112,132), (145,1482), (185,389), (186,403), (193,1351), (376,2), (378,427), (550,140), (925,131), (930,128), (944,1), (1113,1312), (1114,1313), (1141,137), (1292,120), (1293,121), (1294,122), (1295,123), (1296,126), (1297,127), (1298,130), (1299,135), (1300,136), (1303,129), (1350,141), (1593,1595)

X(4) = isogonal conjugate of X(3)
X(4) = isotomic conjugate of X(69)
X(4) = cyclocevian conjugate of X(2)
X(4) = inverse-in-circumcircle of X(186)
X(4) = inverse-in-nine-point-circle of X(403)
X(4) = complement of X(20)
X(4) = anticomplement of X(3)
X(4) = anticomplementary conjugate of X(20)
X(4) = eigencenter of cevian triangle of X(I) for I = 1, 88, 162
X(4) = eigencenter of anticevian triangle of X(I) for I = 1, 44, 513

X(4) = X(I)-Ceva conjugate of X(J) for these (I,J):
(7,196), (27,19), (29,1), (92,281), (107,523), (264,2), (273,278), (275,6), (286,92), (393,459)

X(4) = cevapoint of X(I) and X(J) for these (I,J):
(1,46), (2,193), (3,155), (5,52), (6,25), (11,513), (19,33), (30,113), (34,208), (37,209), (39,211), (51,53), (65,225), (114,511), (115,512), (116,514), (117,515), (118,516), (119,517), (120,518), (121,519), (122,520), (123,521), (124,522), (125,523), (126,524), (127,525), (185,235), (371,372), (487,488)

X(4) = X(I)-cross conjugate of X(J) for these (I,J):
(3,254), (6,2), (19,278), (25,393), (33,281), (51,6), (52,24), (65,1), (113,403), (125,523), (185,3), (193,459), (225,158), (389,54), (397,17), (398,18), (407,225), (427,264), (512,112), (513,108), (523,107)

X(4) = crosspoint of X(I) and X(J) for these (I,J): (2,253), (7,189), (27,286), (92,273)

X(4) = crosssum of X(I) and X(J) for these (I,J):
(4,1075), (6,154), (25,1033), (48,212), (55,198), (56,1035), (71,228), (184,577), (185,417), (216,418)

X(4) = crossdifference of any two points on line X(520)X(647)

X(4) = X(I)-Hirst inverse of X(J) for these (I,J):
(1,243), (2,297), (3,350), (19,242), (21,425), (24,421), (25,419), (27,423), (28,422), (29,415), (420,427), (424,451), (459,460), (470,471), (1249,1503)

X(4) = X(I)-aleph conjugate of X(J) for these (I,J): (1,1047), (29,4)

X(4) = X(I)-beth conjugate of X(J) for these (I,J):
(4,34), (8,40), (29,4), (162,56), (318,318), (811,331)

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