Excellent points in the triangle

The center of the circumference (blue), the center of gravity (green) and the intersection point (red) lie on a straight line

In geometry , the marked points (also: strange points or centers ) of a triangle are primarily the following four points :

the height intersection H (intersection of the heights ),
the center of the circle U (intersection of the perpendicular lines (side symmetries) ),
the center of gravity S (intersection of the bisectors (median lines) ) and
the inscribed center point I (intersection of the bisector (angle symmetry) ).

The first three intersection points (H, U and S) always lie on a straight line , the Euler straight line . On it, in the middle between H and U , lies the center of the Feuerbach circle .

Further points according to the Encyclopedia of Triangle Centers

Triangle with the "classic" excellent points and Euler's straight line

In addition to the four “classic” points of a triangle (center of gravity, circumferential center, incircle center, height intersection), which were already known in antiquity, many other points have been found and investigated in the last few centuries. Clark Kimberling ’s Encyclopedia of Triangle Centers (see web link) lists more than 30,000 (as of January 11, 2019) special points and their previously known properties. The standard designation introduced in this directory, consisting of the letter X and an index, is used today in many treatises on triangular geometry. The following table gives some important examples:

Excellent points in the triangle
Inscribed center	${\ displaystyle X_ {1}}$
main emphasis	${\ displaystyle X_ {2}}$
Circumcenter	${\ displaystyle X_ {3}}$
Intersection point (orthocenter)	${\ displaystyle X_ {4}}$
Center of the Feuerbach circle	${\ displaystyle X_ {5}}$
Lemoine point (symmetry point, Grebe point)	${\ displaystyle X_ {6}}$
Gergonne point	${\ displaystyle X_ {7}}$
Nail point	${\ displaystyle X_ {8}}$
Midpoint	${\ displaystyle X_ {9}}$
Spieker Point (Spieker Center)	${\ displaystyle X_ {10}}$
Feuerbach point (point of contact between the inscribed circle and the Feuerbach circle)	${\ displaystyle X_ {11}}$
1. Fermat point (including the shortest distance to all corner points)	${\ displaystyle X_ {13}}$
2. Fermat point	${\ displaystyle X_ {14}}$
1st isodynamic point	${\ displaystyle X_ {15}}$
2nd isodynamic point	${\ displaystyle X_ {16}}$
1. Napoleon point	${\ displaystyle X_ {17}}$
2. Napoleon point	${\ displaystyle X_ {18}}$
Clawson point	${\ displaystyle X_ {19}}$
Longchamps point	${\ displaystyle X_ {20}}$
Schiffler point	${\ displaystyle X_ {21}}$
Exeter point	${\ displaystyle X_ {22}}$
Bevan point	${\ displaystyle X_ {40}}$
Kosnita point	${\ displaystyle X_ {54}}$
Steiner point	${\ displaystyle X_ {99}}$
Isoperimetric point	${\ displaystyle X_ {175}}$
Point of the same detour	${\ displaystyle X_ {176}}$
1. Vecten point	${\ displaystyle X_ {485}}$
2. Vecten point	${\ displaystyle X_ {486}}$