NAGEL POINT

In the plane of any triangle ABC, let

Ea = the A-excircle,
Eb = the B-excircle,
Ec = the C-excircle.

Let

D = the point where Ea touches side BC,
E = the point where Eb touches side CA,
F = the point where Ec touches side AB.

The lines AD, BE, CF meet in a point, labeled X in the figure. It is called the Nagel point of triangle ABC.

Another construction of the Nagel point follows. Let

d = the point half way around the perimeter of ABC, starting at A,
e = the point half way around the perimeter of ABC, starting at B,
f = the point half way around the perimeter of ABC, starting at C,

The lines Ad, Be, Cf meet in the Nagel point.

C. H. von Nagel
Triangle Centers
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