## X(24) (PERSPECTOR OF ABC AND ORTHIC-OF-ORTHIC 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           sec A cos 2A : sec B cos 2B : sec C cos 2C
= sec A - 2 cos A : sec B - 2 cos B : sec C - 2 cos C

Barycentrics     tan A cos 2A : tan B cos 2B : tan C cos 2C
= tan A - sin 2A : tan A - sin 2B : tan C - sin 2C

Constructed as indicated by the name; also X(24) = X(56)-of-the-tangential triangle if ABC is acute.

X(24) lies on these lines:
1,1061    2,3    6,54    32,232    33,35    34,36    49,568    51,578    52,1147    56,1870    64,74    96,847    98,1289    107,1093    108,915    110,155    154,1181    182,1843    183,1235    184,389    185,1495    242,1602    254,393    264,1078    511,1092    573,1474    602,1395    944,1610    1063,1775    1112,1511    1192,1511    1324,1603    1385,1829

X(24) is the {X(3),X(4)}-harmonic conjugate of X(378).

X(24) = reflection of X(4) in X(235)
X(24) = isogonal conjugate of X(68)
X(24) = inverse-in-circumcircle of X(403)
X(24) = X(249)-Ceva conjugate of X(112)
X(24) = X(52)-cross conjugate of X(4)
X(24) = crosspoint of X(107) and X(250)
X(24) = crosssum of X(I) and X(J) for these (I,J): (6,161), (125,520), (637,638)
X(24) = X(4)-Hirst inverse of X(421)

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