From the feet 
,
, and 
of each altitude of a triangle 
, draw lines 
, 
, 
perpendicular
to the adjacent sides, as illustrated above. Then the points 
, 
,
, 
, 
, and 
are concyclic, and the circle
passing through these points is called the Taylor circle. Here, 
, 
, and 
are antiparallel to the sides 
, 
,
and 
,
respectively.
Furthermore, the figures 
and 
are similar, where 
is the orthocenter of 
, 
is parallel to 
, and 
bisects 
and 
.
If 
is the circumradius
of the reference triangle, then
 |
(1)
|
Also, if the triangle is acute, this is equal to
 |
(2)
|
(Johnson 1929, p. 277).
The Taylor circle has circle function
 |
(3)
|
which corresponds to Kimberling center 
. The center of the circle has
trilinear function
 |
(4)
|
which is Kimberling center 
, is the center of the Spieker
circle of the orthic triangle of the reference
triangle (Johnson 1929, p. 277), and is called the Taylor
center.
The radius of the Taylor circle is given by
 |
(5)
|
No notable triangle centers lie on the Taylor circle.
The Taylor circle is a Tucker circle with parameter
 |
(6)
|
|
|
|
|
There are a number of remarkable properties satisfied by the figure obtained in the construction of the Taylor circle:
1. The feet of the perpendiculars from a given altitude foot are concyclic with the opposite vertex.
2. The two feet of the perpendiculars which are closest to a given vertex are concyclic with the feet of the altitudes on the corresponding sides.
3. The two feet of the perpendiculars which are closest to a give vertex are concyclic with that vertex and with the intersection of the perpendiculars.
4. The three circles through the orthocenter and the feet of the perpendiculars on a given side intersect pairwise along the altitudes.
The first three of these follow from that fact that 
is equivalent to 
lying on the circle with diameter 
, and the fourth follows from the concurrence of the pairwise
radical axes of three circles (the three circles being two of the circles through
the orthocenter and the feet of the perpendiculars on a given side, and the Taylor
circle).
