X(849) (4th HATZIPOLAKIS-YIU POINT)

 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.

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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            [a/(b + c)]2 : [b/(c + a)]2 : [c/(a + b)]2
Barycentrics    a[a/(b + c)]2 : b[b/(c + a)]2 : c[c/(a + b)]2

Let D denote the circumcircle of triangle ABC. Let DA be the circle tangent to sideline BC and tangent to D at A. Let Ba = AC∩DA and Ca = AB∩DA, and define Cb, Ca and Ac, Ab cyclically. Define A' = CbAb∩AcBc, and define B' and C' cyclically. Then triangle A'B'C' is homothetic to ABC, and X(849) is the center of homothety. See A. Hatzipolakis and P. Yiu, Hyacinthos #2056-2070, December, 2000.

X(849) lies on these lines: 32,163    36,58    110,595    249,1110    741,827    757,763

X(849) = isogonal conjugate of X(1089)
X(849) = X(249)-Ceva conjugate of X(163)
X(849) = crosspoint of X(58) and X(501)
X(849) = crosssum of X(10) and X(502)

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