## X(65) (ORTHOCENTER OF THE INTOUCH TRIANGLE)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
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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.

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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 C + cos A : cos A + cos B
= (b + c)/(b + c - a) : (c + a)/(c + a - b) : (a + b)/(a + b - c)
= sin(A/2) cos(B/2 - C/2) : sin(B/2) cos(C/2 - A/2) : sin(C/2) cos(A/2 - B/2)

Barycentrics    a(b + c)/(b + c - a) : b(c + a)/(c + a - b) : c(a + b)/(a + b - c)

X(65) is the perspector of ABC and the extangents triangle.

X(65) lies on these lines:
1,3    2,959    4,158    6,19    7,8    10,12    11,117    29,296    31,1104    33,64    37,71    41,910    42,73    44,374    58,109    63,958    68,91    74,108    77,969    79,80    81,961    110,229    169,218    172,248    224,1004    225,407    243,412    257,894    278,387    279,1002    386,994    409,1098    474,997    497,938    516,950    519,553    604,1100    651,895    1039,1041    1061,1063

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

X(65) = reflection of X(I) in X(J) for these (I,J): (1,942), (72,10)
X(65) = isogonal conjugate of X(21)
X(65) = isotomic conjugate of X(314)
X(65) = anticomplement of X(960)
X(65) = X(I)-Ceva conjugate of X(J) for these (I,J): (1,73), (4,225), (7,226), (10,227), (109,513), (226,37)
X(65) = X(42)-cross conjugate of X(37)
X(65) = crosspoint of X(I) and X(J) for these (I,J): (1,4), (7,57)
X(65) = crosssum of X(I) and X(J) for these (I,J): (1,3), (9,55), (56,1394)
X(65) = crossdifference of any two points on line X(521)X(650)
X(65) = X(1284)-Hirst inverse of X(1400)
X(65) = X(I)-beth conjugate of X(J) for these (I,J):
(1,65), (8,72), (10,10), (65,1042), (80,65), (100,65), (101,213), (291,65), (668,65), (1018,65)

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