## X(5) (NINE-POINT CENTER)

 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 - C) : cos(C - A) : cos(A - B)
= f(A,B,C) : f(B,C,A) : f(C,A,B), where f(A,B,C) = cos A + 2 cos B cos C
= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = cos A - 2 sin B sin C
= h(a,b,c) : h(b,c,a): h(c,a,b), where h(a,b,c) = bc[a2(b2 + c2) - (b2 - c2)2]

Barycentrics    a cos(B - C) : b cos(C - A) : c cos(A - B)
= h(a,b,c) : h(b,c,a) : h(c,a,b), where h(a,b,c) = a2(b2 + c2) - (b2 - c2)2

X(5) is the center of the nine-point circle. Euler showed in that this circle passes through the midpoints of the sides of ABC and the feet of the altitudes of ABC, hence six of the nine points. The other three are the midpoints of segments A-to-X(4), B-to-X(4), C-to-X(4). The radius of the nine-point circle is one-half the circumradius.

Dan Pedoe, Circles: A Mathematical View, Mathematical Association of America, 1995.

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

X(5) = midpoint of X(I) and X(J) for these (I,J):
(1,355), (2,381), (3,4), (11,119), (20,382), (68,155), (110,265), (113,125), (114,115), (116,118), (117,124), (122,133), (127,132), (128,137), (129,130), (131,136)

X(5) = reflection of X(I) in X(J) for these (I,J): (2,547), (3,140), (4,546), (20,548), (52,143), (549,2), (550,3), (1263,137), (1353,6), (1385,1125), (1483,1), (1484,11)

X(5) = isogonal conjugate of X(54)
X(5) = isotomic conjugate of X(95)
X(5) = inverse-in-circumcircle of X(2070)
X(5) = inverse-in-orthocentroidal-circle of X(3)
X(5) = complement of X(3)
X(5) = anticomplement of X(140)
X(5) = complementary conjugate of X(3)
X(5) = eigencenter of anticevian triangle of X(523)

X(5) = X(I)-Ceva conjugate of X(J) for these (I,J):
(2,216), (4,52), (110,523), (264,324), (265,30), (311,343), (324,53)

X(5) = cevapoint of X(I) and X(J) for these (I,J): (3,195), (51,216)

X(5) = X(I)-cross conjugate of X(J) for these (I,J): (51,53), (216,343), (233,2)
X(5) = crosspoint of X(I) and X(J) for these (I,J): (2,264), (311,324)
X(5) = crosssum of X(I) and X(J) for these (I,J): (3,1147), (6,184)
X(5) = crossdifference of any two points on line X(50)X(647)
X(5) = X(1)-aleph conjugate of X(1048)

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