Triangle centre

2020 Mathematics Subject Classification: Primary: 51M15 [MSN][ZBL]

Given a triangle $A_1A_2A_3$, a triangle centre is a point dependent on the three vertices of the triangle in a symmetric way. Classical examples are:

the centroid (i.e. the centre of mass), the common intersection point of the three medians (see Median (of a triangle));

the incentre, the common intersection point of the three bisectrices (see Bisectrix) and hence the centre of the incircle (see Plane trigonometry);

the circumcentre, the centre of the circumcircle (see Plane trigonometry);

the orthocentre, the common intersection point of the three altitude lines (see Plane trigonometry);

the Gergonne point, the common intersection point of the lines joining the vertices with the opposite tangent points of the incircle;

the Fermat point (also called the Torricelli point or first isogonic centre), the point $X$ that minimizes the sum of the distances $|A_1X|+|A_2X|+|A_3X|$;

the Grebe point (also called the Lemoine point or symmedian point), the common intersection point of the three symmedians (the symmedian through $A_i$ is the isogonal line of the median through $A_i$, see Isogonal);

the Nagel point, the common intersection point of the lines joining the vertices with the centre points of the corresponding excircles (see Plane trigonometry).

the nine-point centre, the centre of the nine-point circle.

In [a1], $400$ different triangle centres are described.

The Nagel point is the isotomic conjugate of the Gergonne point, and the symmedean point is the isogonal conjugate of the centroid (see Isogonal for both notions of "conjugacy" ).

References

[a1]	C. Kimberling, "Triangle centres and central triangles" Congr. Numer. , 129 (1998) pp. 1–285
[a2]	R.A. Johnson, "Modern geometry" , Houghton–Mifflin (1929)

Comments

The Euler line contains some of the classical centres: the centroid, the orthocentre, the circumcentre and the nine-point centre.

References

[b1]	H. S. M. Coxeter, Samuel L. Greitzer, "Geometry Revisited" New Mathematical Library 19 Mathematical Association of America (1967) ISBN 0883856190 Zbl 0166.16402