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See K005 and K397 for a similar property

A1, B1, C1 reflections of A, B, C

A1*, B1*, C1* their isogonal conjugates

feet of the trilinear polar of X(5)

Let A1, B1, C1 be the reflections of A, B, C in the sidelines of triangle ABC. For any point M, denote by Ma the intersection of the lines MA1 and BC. Define Mb and Mc similarly. These points Ma, Mb, Mc form a triangle perspective to ABC if and only if M lies on the Neuberg cubic. This property is equivalent to locus property 1 in the Neuberg cubic page.

These points Ma, Mb, Mc are collinear if and only if M lies on K216. Hence, K216 is a special case of Grassmann cubic. See CL041.

K216 is an isogonal nK with root X(5), the nine point center.


Consider the pencil of circles generated with the polar circle (P) and the circle with diameter HK.

Their radical axis (L) is the trilinear polar of X(264).

This pencil contains two remarkable circles related with K216, namely :

• (C) with center Ω = X(6146) in green,

• (c) with center the reflection ω of X(4) about X(6146), in blue.

ω is unlisted in the current edition of ETC.

It lies on the lines {X3,X70},{X4,X6},{X5,X1614},{X22,X68},{X24,X1899},{X54,X427},{X74,X550},{X96,X98},{X156,X2072},{X184,X1594},{X858,X1147}.

Its SEARCH number is 11.6640123927717.


K216 is then the locus of M such that :

• the pedal circle of M and M* is orthogonal to (C),

• M and M* are conjugated in (c).

The figure shows :

• the pedal circles (A1), (B1), (C1) of A1, B1, C1 (dashed green),

• the polar lines (La), (Lb), (Lc) of A1, B1, C1 in (c) (dashed blue) passing through the isogonal conjugates of A1, B1, C1.