## X(1319) (BEVAN-SCHRÖDER POINT)

 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           f(a,b,c) : f(b,c,a) : f(c,a,b), where f(a,b,c) = (b + c - 2a)/ (b + c - a)
= g(A,B,C) : g(B,C,A) : g(C,A,B), where g(A,B,C) = 2 - 2 cos A - cos B - cos C (Peter J. C. Moses)

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

Let X'Y'Z' be the pedal triangle of the Bevan point, W = X(40); then X(1319) is the point, other than W, in which the circles AWX', BWY', CWZ' concur. (Floor van Lamoen, Hyacinthos #6321, 6352).

X(1319) lies on these lines:
1,3    11,515    12,1125    37,604    44,1317    48,1108    59,518    73,1104    77,1122    106,1168    108,953    210,956    214,519    226,535    355,499    392,993    513,663    529,908    840,934    910,1055    961,1255

X(1319) is the {X(1),X(56)}-harmonic conjugate of X(65).

X(1319) = midpoint of X(1) and X(36)
X(1319) = reflection of X(1155) in X(36)
X(1319) = isogonal conjugate of X(1320)
X(1319) = inverse-in-circumcircle of X(56)
X(1319) = inverse-in-incircle of X(65)
X(1319) = cevapoint of X(902) and X(1404)
X(1319) = crosspoint of X(1) and X(104)
X(1319) = crosssum of X(1) and X(517)
X(1319) = crossdifference of any two points on line X(9)X(650)

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