the lester circle and others

On June Lester's page she describes the Lester circle. This passes through the well-known triangle centres
the circumcentre, the nine-point centre, and the Fermat points. Her proof involves quite complicated algebraic
manipluation. I noticed that there is a simpler proof by inversive geometry (given below). I now realise that
this proof was discovered by Conway - see hyacinthos newsgroup #1284 etc. Conway also observed that
the same proof shows that there is a circle through the first two points, the symmedian point and the centre
of the Kiepert hyperbola. Both circles are orthogonal to the orthocentroidal circle (aka the Guinard ring).

In fact, there is a family of 24 such circles (and two lines). There are also families associated with the circumcircle,
the nine-point circle and the Brocard circle. The Parry circle is remarkable in that it occurs in the families of the
circumcircle and the Brocard circle - so contains many triangle centres.

Some special triangle centres

There are many points associated with a triangle, such as the incentre and circumcentre.
A formal definition and a comprehensive list appear in the Kimberling's encyclopedia

X(1)	incentre
X(2)	centroid
X(3)	circumcentre
X(4)	orthocentre
X(5)	centre of nine-point circle
X(6)	symmedian point
X(13),X(14)	Fermat points
X(15),X(16)	isodynamic points
X(23)	far-out point
X(115)	centre of the kiepert hyperbola

some special lines and circles

Many lines and circles contain several of these centres. An extensive survey appears in
Edward Brisse's list

e	Euler line
f	Fermat axis
b	Brocard axis
O	orthocentroidal circle or guinard ring
C	circumcircle
N	nine-point circle
B	Brocard circle

The three lines e,f,b coincide if and only if the triangle is isosceles.

two basic results from inversive geometry

lemma 1
(1) If A,B are inverse with respect to a circle C,
then any line or circle through A and B is orthogonal to C.
(2) If A,B are not inverse with respect to a circle C,
then there is a unique circle or line through A and B orthogonal to C.

proof

lemma 2
If {A,B}, {C,D} are pairs of points inverse with respect to a circle C,
then A,B,C,D lie on a circle or line which is orthogonal to C.
If {A,B}≠{C,D}, then the circle or line is unique.

proof
Since each pair consists of points inverse with respect to C, the pairs
are identical or disjoint.
In the former case, infinitely many circles exist by lemma 1(1).
Otherwise, the unique circle or line through A and C orthogonal to C
guaranteed by lemma 1(2) passes through B and D by orthogonality.

To use this information, we need pairs of points inverse with respect to a circle.

table of inverse pairs

From the encyclopedia, there are many pairs of centres which are inverse with respect to well-known circles.
We can summarize the information as follows:

orthocentroidal circle			circumcircle			Brocard circle
O1	e	X(3),X(5)	C1	2	X(1),X(36)	B1	148	X(2),X(110)
O2	f	X(6),X(115)	C2	e	X(2),X(23)	B2	b	X(15),X(16)
O3	f	X(13),X(14)	C3	e	X(4),X(186)	B3	b	X(32),X(39)
O4	e	X(25),X(427)	C4	b	X(6),X(187)	B4	b	X(50),X(586)
O5	e	X(27),X(469)	C5	b	X(15),X(16)	B5	b	X(58),X(386)
O6	e	X(297),X(458)	C6	e	X(22),X(858)	B6	b	X(61),X(62)
O7	e	X(378),X(403)	C7	e	X(24),X(403)	B7	3895	X(111),X(353)
O8	e	X(379),X(857)	C8	e	X(25),X(468)	B8	148	X(125),X(184)
O9	e	X(382),X(546)	C9	2	X(35),X(484)	B9	b	X(187),X(574)
O10	e	X(383),X(1080)	C10	311	X(352),X(353)	B10	b	X(216),X(577)
O11	e	X(405),X(442)	C11	336	X(667),X(1083)	B11	b	X(284),X(579)
O12	e	X(406),X(475)				B12	b	X(371),X(372)
O13	e	X(470),X(471)				B13	b	X(389),X(578)
O14	e	X(472),X(473)				B14	b	X(500),X(582)
			nine-point circle			B15	b	X(567),X(568)
			N1	e	X(2),X(858)	B16	b	X(52),X(569)
			N2	e	X(4),X(403)	B17	b	X(570),X(571)
			N3	510	X(141),X(625)	B18	b	X(572),X(573)
			N4	e	X(427),X(468)	B19	b	X(575),X(576)
			N5	510	X(623),X(624)	B20	b	X(580),X(581)
						B21	b	X(583),X(584)

In the table, the letters e,f,b indicate that the points lie on the Euler line or on the Fermat or Brocard axis.
A number in this column is the number of a line in Brisse's list.

We can now describe the various families :

• the orthocentroidal family
  If ΔABC is not isosceles, then the pairs {{Om,On} : m≠n} define a family of 24 distinct circles,
  and two lines, each orthogonal to the orthocentroidal circle and containing 4 named centers.
  proof
  By Lemma 2, each pair defines a unique line or circle orthogonal to the orthocentroidal circle.
  If {m,n}={2,3}, we get the Fermat axis. If m,n≠2,3, we get the Euler line. Otherwise, we get a
  circle since we have two points on each of these distinct lines. The circles are distinct since any
  three pairs would contain four points on one of these lines.
  Notes.
  The family contains the Lester circle {O1,O3} and the circle noted by Conway {O1,O2}.
• the nine-point family
  If ΔABC is not isosceles, then the pairs {{Nm,Nn} : m≠n} define a family of 6 distinct circles,
  and two lines, each orthogonal to the nine-point circle and containing 4 named centers.
  The proof is exactly as above. The lines are the Brocard axis and Brisse's line 510.
• the Brocard family
  If ΔABC is not isosceles, then the pairs {{Bm,Bn} : m≠n} define a family of 54 circles,
  and two lines, each orthogonal to the Brocard circle and containing 4 named centers.
  The usual proof shows that each pair gives a line or circle orthogonal to the Brocard circle.
  There are 210 pairs, but 153 define the Brocard axis (m,n≠1,7,8), and any two of B1, B2, B7
  define the Parry circle. Thus we have at most 56 distinct objects. The lines are the Brocard axis
  and Brisse's line 148.
  Here, we cannot assert that the circles are distinct. Briss's lists are offline, but should show that
  they are. We can observe that, as above, the 36 circles with m,n≠7 are distinct, and that the 18
  circles with m,n≠1,8 are also distinct. Of course, each of these subfamilies includes the Parry circle.
  The Parry circle occurs again as {B1,B7}. The last circle is then {B7,B8}.
• the circumcircle family
  If ΔABC is not isosceles, then the pairs {{Cm,Cn} : m≠n} define a family of 41 circles,
  and three lines, each orthogonal to the circumcircle and containing 4 named centers.
  The usual proof shows that each pair gives a line or circle orthogonal to the circumcircle.
  The lines are the Brocard axis, the Euler line and Brisse's line 2. Since 5 pairs lie on the Euler line,
  and two on each of the others, 12 of the 55 pairs lead to lines. Any two of C2, C5, C10 define
  the Parry circle, so we have at most 41 distinct circles.

Notes
(1) The Parry circle appears in the last two families - this shows that it contains the points X(2),X(15),X(16),X(23),
X(110),X(111),X(352),X(353). These are all the named centers known to lie on the circle.
(2) The points X(15) and X(16) are inverse for both the circumcircle and the Brocard circle, so these are disjoint.
Also, inversion in a circle with centre X(15) or X(16) maps the circumcircle and Brocard circle to concentric circles.
(3) Any circle through a pair is orthogonal to the corresponding circle, so we get a large number of circles each
orthogonal to one of the named circles and through at least three named centers - e.g. for n≠15,16, if X(n) is not
on the Brocard axis, then the circle through X(15),X(16),X(n) is orthogonal to the circumcircle (and to the Brocard circle).
Taking the case of the circumcircle, this proves the orthogonality of 511 of the 513 circles listed by Brisse as being
orthogonal to the circumcircle.

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