The Triangle of Centers (full version)

This picture is the best thing we have as a map of the structures in Triangle Geometry. One can spend hours looking at this. There is a lot of information here. This is John Conwy's organization of the strong points in the plane of the triangle.

All the points in this picture are strong points. Here is a Geometer's Sketchpad document that defines all these points from G,O and K, chosing K randomly as a point inside the Guinard Shield, the circle defined on GH.

In addition to points and lines, the circumcircle, the Keipert and Jerabek hyperbolas, the Steiner ellipse and the Guinard Sheld circle are shown, as well as (very lightly, many of the coaxal family of circles with centers on the Euler line.

The Guinard shield plays an important role in The Triangle Book.

The four bold lines through the centroid are the Euler line, the symmedian track, the Steiner track, and the Tarry track. The last two are the Euler line and the symmedian track for the Brocard image triangle, which also plays a very important role in The Triangle Book.

An index of notation is below the picture.

Index of Notation

Operations

notation	name	comments	rule
t	isotomic conjugate	affine invariant , takes line at infinity to Steiner ellipse	:y: -> :1/y:
g	isogonic conjugate	takes line at infity to circumcircle	:y: -> :b2/y:
p	protacted	= gt, projective, takes point on Steiner ellipse to one on circumcircle	:y: -> :b2y:
r	retracted	= tg, projective , takes point on circumcircle to one on Steiner ellipse	:y: -> :y/b2:
m	medial	same point wrt median triangle, affine invariant	:y: -> :z+x:
d	dilated	same point wrt antimedial triangle, affine invariant	:y: -> :z+x-y:
v	inverse	inverse to circumcircle
o	opposite	reflect through circumcenter	:y: -> :S2 y - 2b2SB: y normalized
n	negated	reflect through centroid	:y: -> :3y - 2: y normalized
i	infra	same point wrt Brocard image triangle
h	hyper	same point wrt triangle for wich ABC is the Brocard image triangle
a, b, c		Globe operation: inverses in the three Apollonian circles
e, w	east and west	Globe operation; go 1/3 of the way around the globe on your lattitude line.

 

Here S is twice the area and S_B = (c² + a² – b²)/2

Catalog of Points

John feels that most interesting points can be created as derivatives (using the above operations) from these points.

Semi-strong points are points, such as the isodynamics, that come in two versions. As a group they are strong, and always define a strong line.

notation	name	comments	y coordinate
A,B,C	triangle vertices
D = tH	Desmon, = tH, the isotomic orthocenter
E	Eulerian crossing point, where orthic axis crosses the Euler line
F; Fn, Fs; Fo	Focus of Kiepert parabola; Fermat points (normal and switched); Feuerback point
G; Go	centroid; Gergonne
H	orthocenter
In, Is Io	Isodynamic points (normal and switched); Incenter
J; Jo	Center of Jerabek hyperbola; Jerabek point
K	symmedian point
L	deLongchamps point
M = mS; Mo	Center of the Kiepert hyperbols = medial Steiner point; Mittenpunkt
N Nn, Ns No	nine point center , Napoleon points Nagel point
O	circumcenter
Po=pK, P2+, P2– ; Pn, Ps	pK, and the two Lucas points (called protracted (?) points); Pythagorean points (aka Vecten points)
Qo, Q2+, Q2– Qo	not sure what their name is; Eulerian correspondence point.		SB2, SB2+ SB2–
R = rG = tK; Ro	not sure what John calls it "retro point" perhaps; retro-incenter = isotomic incenter
S So	Steiner point Spieker point
T	Tarry point
U	Umbo point (Greek word for center of a shield)
V	Vertex of Kiepert parabola
W; Wo	Taylor's Wheelcenter Clawson point
X = vK	Crossing point, where Brocard Meridian Line and Lemoine Lattitude Line meet = inverse of symmedian in circumcircle
Y	Far out point; not sure John's name for it; I think he quixotically calls it the "yonder" point
Z	Zeeman-Gossard perspector