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30 Pythagoras 63, June, 2006, pp. 30-37
A generalisation of the Spieker circle and Nagel line
Michael de Villiers
School of Science, Mathematics, & Technology Education, University of KwaZulu-Natal
profmd@mweb.co.za
Introduction
Many a famous mathematician and scientist have
described how their first encounter with Euclidean
geometry was the defining moment in their future
careers. Some of the most well known are probably
Isac Newton and Albert Einstein. Often these
encounters in early adolescence have been
poetically described as passionate love affairs. For
example, the mathematician Howard Eves
describes his personal experience as follows:
“...Euclid's Elements ... I leafed through the book,
and found that, from a small handful of
assumptions ... all the rest apparently followed by
pure reasoning ... The experience had all the
aspects of a romance. It was love at first sight. I
soon realized I had in my hands perhaps the most
seductive book ever written. I fell head over heels
in love with the goddess Mathesis ... As the years
have gone by I have aged, but Mathesis has
remained as young and beautiful as ever” (in
Anthony, 1994: xvi-xvii)
Perhaps noteworthy is that very few famous
mathematicians and scientists have ever mentioned
arithmetic or school algebra as having been as
influential as geometry in attracting them to
mathematics. One of the reasons may be the
algorithmic nature of high school algebra as
pointed out by Howard Eves as follows: “ ... I still
think that geometry is the high school student's
gateway to mathematics. It's not algebra, because
high school algebra is just a collection of rules and
procedures to be memorized” (in Anthony, 1994:
xvii). Moreover, the fundamental mathematical
idea of proof, and that of a deductive structure and
of logical reasoning, is usually introduced and
developed largely within high school geometry,
and hardly at all in algebra. Though elementary
number theory and algebra can provide exciting
opportunities for some conjecturing and proof, it is
unfortunately not common practise in high school.
At present, it is mainly geometry that provides a
challenging, non-routine context for creative proof
that requires learners to explore and discover the
logical links between premises and conclusions.
The current reduction of Euclidean geometry
from the new South African school curriculum at
the General Education and Training (GET) and
Further Education and Training (FET) levels has
been largely motivated by the need to introduce
some more contemporary topics. Some of these are
cartesian and transformation geometry, as well as a
little non-Euclidean geometry such as spherical
geometry, taxi-cab geometry and fractal geometry.
However, it would seem disastrous for the future
development of mathematicians and scientists in
our country to argue, as some do, for the complete
removal of Euclidean geometry from the
curriculum. Often the argument seems a purely
political one: learners find geometry difficult
compared to algebra; we have to improve the pass
rate; so let's get rid of geometry!
Of course, the problem of geometry education
is a very complex one, and is not one that I will
attempt to address in this article, though some of
my mathematics education research and thoughts
in this regard appear in De Villiers (1997). It is
also not a problem limited to our country, but is
fairly international. Suffice to say that ignoring the
problem will not solve it, but that it has to be faced
head on, and will require the concerted, combined
efforts of mathematicians, mathematics educators,
teachers and researchers.
This article instead modestly aims to acquaint
the reader with some results from 17th and 19th
century geometry, and to combat the perspective
that geometry is dead by showing that new
discoveries can and are still being made.
Specifically it will discuss a possibly new
generalisation of the Spieker circle and the
associated Nagel line, which is parallel to that of
the generalisation of the nine-point circle and Euler
line discussed in De Villiers (2005). Not only
should these results be accessible to a fair number
of undergraduate students, prospective and
practising high school teachers, but also to the
more mathematically talented high school learner.
Unlike cutting edge research in other areas of
mathematics, the results are relatively easy to
understand and appreciate, even without proof,
because of their visual nature.
Apart from the remarkable concurrencies of the
medians, altitudes and perpendicular bisectors of a
Michael de Villiers
31
triangle mentioned in De Villiers (2005), there is a
fourth concurrency theorem mentioned in a few
South African textbooks, namely:
The angle bisectors of the angles of a
triangle are concurrent at the incentre,
which is the centre of the inscribed circle
of the triangle (see Figure 1).
Figure 1: Incentre
Nagel point
Many mathematics teachers are not aware that
there are many famous special centres associated
with the triangle, and not only the four, i.e. the
centroid, orthocentre, circumcentre and incentre,
normally mentioned in textbooks. In fact, Clark
Kimberling's two websites are worth a visit, where
over 1000 special centres are associated with the
triangle (see Kimberling)! Antonio Gutierrez's site
also provides some beautiful, draggable dynamic
geometry sketches of some of the more famous
triangle centres (see Gutierrez).
One such notable point is the Nagel point,
which is the point of concurrency of the lines from
the vertices of a triangle to the points on the
opposite sides where they are touched by the
escribed circles (see Figure 2). This interesting
point is named after its discoverer, the German
mathematician Christian Heinrich von Nagel
(1803-1882) and some biographical information
about him can be obtained from:
http://faculty.evansville.edu/ck6/bstud/nagel.html
Pascal's theorem
The French philosopher and mathematician Blaise
Pascal (1623-1662) discovered and proved the
following remarkable theorem at the age of
sixteen: All six vertices of a hexagon lie on a
conic, if and only if, the intersections of the three
pairs of opposite sides are collinear (lie on a
straight line) – see Figure 3 on following page.
This is one of the first entirely projective theorems
discovered and proved, and does not involve any
measurement of sides or angles. Note that when
the opposite sides of the inscribed hexagon are
parallel they are assumed to meet at infinity, and
all points at infinity are assumed to lie on the line
at infinity. Pascal's proof has unfortunately not
survived, but he probably used classical Euclidean
geometry, and not modern projective methods.
y
y
x
x
I
Figure 2: Nagel point
N
A generalisation of the Spieker circle and Nagel line
32
Spieker circle and Nagel line
The discovery of the nine-point circle and the
associated Euler-line has often been described as
one of the crowning glories of post-Greek
synthetic geometry (see De Villiers, 2005 for more
details). However, less well known seems to be an
interesting analogue or parallel result involving the
Spieker circle and the Nagel line. The Spieker
circle is named after Theodor Spieker whose 1890
geometry book Lehrbuch der ebenen Geometrie
was one of the books that greatly inspired the
young Einstein (see Pyenson, 1985). The rather
remarkable parallelism between the nine-point
circle and Euler line on the one hand, and that of
the Spieker circle and Nagel line on the other hand,
is contrasted in the table below, and illustrated in
Figure 4. (The reader is reminded that the median
triangle is the one formed by the midpoints of the
sides of a triangle.)
The Spieker circle
The nine-point circle
N
PG
OG S
H
I
A
B C
A
B C
Figure 4: Nine-point & Spieker circles
Figure 3: Pascal’s theorem
Michael de Villiers
33
The nine-point circle is the circumcircle of
ABC's median triangle and has radius half
that of circumcircle of ABC.
The Spieker circle is the incircle of ABC's
median triangle and has radius half that of
incircle of ABC.
The circumcentre (O), centroid (G) &
orthocentre (H) of any triangle ABC are
collinear (Euler line), GH = 2GO and the
midpoint of OH is the centre of the nine-
point circle (P) so that HP = 3 PG.
The incentre (I), centroid (G) & Nagel point
(N) of any triangle are collinear (Nagel line),
GN = 2GI and the midpoint of IN is the
centre of the Spieker circle (S) so that NS = 3
SG.
The nine-point circle cuts the sides of ABC
where the extensions of the altitudes through
the orthocentre meet the sides of ABC.
The Spieker circle touches the sides of the
median triangle where they meet the lines
from the Nagel point to the vertices of ABC.
The nine-point circle passes through the
midpoints of the segments from the
orthocentre to the vertices of the triangle.
The Spieker circle touches the sides of the
triangle whose vertices are the midpoints of
the segments from the Nagel point to the
vertices of ABC.
The above observations are not new, and appear
together with proofs in Coolidge (1971) and
Honsberger (1995). More generally, this is an
example of a limited, but an interesting kind of
duality not only between the incircles (and
escribed circles) and circumcircles of triangles and
other polygons, but also between the concepts of
side and angle, equal and perpendicular diagonals
(e.g. for quadrilaterals), etc. This limited duality or
analogy is explored fairly extensively in De
Villiers (1996) and has been useful in formulating
and discovering several new results (see for
example De Villiers, 2000).
Having recently rediscovered a generalisation of
the nine-point circle to a nine-point conic and an
associated generalisation of the Euler line (De
Villiers, 2005), the author wondered how one
could generalise the Spieker circle (and the Nagel
line) in a similar way. The following is the result
of that investigation.
Let us first prove the following useful Lemma that
will come in handy a little later. The first algebraic
proof is my own original one while the second
geometric one was kindly sent to me by Michael
Fox from Leamington Spa, Warwickshire, UK.
Figure 5: Lemma
A generalisation of the Spieker circle and Nagel line
34
Lemma
Given A', B', C' as the images of any three points
A, B, C after a half-turn about O, then the six
points A', B', C', A, B, and C lie on a conic.
Algebraic proof
Place O at the origin so that the general conic
equation we need to determine reduces to
ax2+2hxy +by2+c=0. Divide through by c,
so that only three unknowns now need to be
determined. Due to the symmetry of the conic
equation it follows that if (x
1
,y
1
) satisfies the
equation so does its image (
−
x
1
,
−
y
1
)under a half-
turn. Therefore, only three points are needed to
find the conic, one from each symmetric pair, e.g.
A or A', B or B', C or C'.
Geometric proof
Consider the hexagon AB'CC'BA' shown in Figure
5. The image of AB' is A'B, therefore AB' // A'B;
similarly BC' // B'C. Thus BL'B'L is a
parallelogram, and its diagonals bisect each other.
But the midpoint of BB' is O, hence LOL' is a
straight line. These are the intersections of the
opposite sides of the hexagon, so by the converse
of Pascal's theorem, the vertices A, B, C, A', B', C'
lie on a conic.
Spieker conic
Given A'B'C' as the median triangle of a triangle
ABC, and A'D, B'E and C'F are three lines
concurrent at N. Let L, J and K be the respective
midpoints of A'N, B'N, and C'N, and X, Y and Z be
the midpoints of the sides of A'B'C' as shown in
Figure 6. For purposes of clarity, an enlargement
of the median triangle and only the relevant points
are shown in the bottom part of Figure 6 (see
following page.)
Since both XK and LZ are parallel and
equal to half B'N, it follows that XKLZ is a
parallelogram. Similarly JXYL and ZJKY are
parallelograms. Let S be the common midpoint of
the respective diagonals XL, YJ, and ZK of these
parallelograms. Further let the intersections of AN,
BN and CN with the sides of the median triangle be
P, Q, and R, and their respective reflections
through S be P', Q', and R'. If a conic is now drawn
through any five of P, Q, R, P', Q', and R', then the
conic cuts through the sixth point, and is inscribed
in the median triangle (as well as the triangle
obtained from the median triangle through a half-
turn around S).
Proof
Since P, Q, R, P', Q', and R' are symmetrically
placed around S by construction, it immediately
follows from the preceding lemma that all six
points lie on the same conic. Furthermore, it is
obvious that projecting the lines A'D, B'E and C'F
onto the altitudes of the median triangle, reduces
the conic to the Spieker circle. Since the Spieker
circle is inscribed in the median triangle (as well as
its half-turn around S), and since any conic and
tangents remain a conic and tangents under
projection, it therefore follows that the general
Spieker conic is also inscribed in both triangles.
Nagel line generalisation
Given the above configuration for any triangle
ABC, then the centre of the Spieker conic (S), the
centroid of ABC (G) and N are collinear, and NS =
3 SG.
Proof
The projection of the Spieker conic onto the
Spieker circle, also projects S onto the the centre of
the Spieker circle, and the point N onto the Nagel
point, and since collinearity is preserved under
projection, S, G and N are collinear. However,
since ratios of segments are not necessarily
invariant under projection, this is not sufficient to
prove NS = 3 SG.
However, this follows directly from the nine-
point conic result and associated Euler
generalisation discussed in De Villiers (2005). In
Figure 6, the nine-point conic result implies that X,
K, Y, E, L, Z, F, J, and D also lie on a conic, and
that it has the same centre S as the Spieker conic.
Hence, the Euler line corollary of this inscribed
nine-point Spieker conic, directly proves the Nagel
generalisation above, so that the centre S of this
nine-point conic, the centroid G of ABC and the
point of concurrency N, are collinear, and NS = 3
SG.
Concluding comments
It is hoped that this article has to some extent
expelled the myth that the ancient Greeks already
discovered and proved everything there is to find
and prove in geometry. Apart from these results
being easily accessible to undergraduate students,
they are probably also within reach of talented high
school students, particularly those at the level of
the Third Round of the Harmony SA Mathematics
Olympiad.
Michael de Villiers
35
Figure 6: Spieker conic
A generalisation of the Spieker circle and Nagel line
36
Moreover, this article has hopefully also
demonstrated that possible new geometric
discoveries such as the nine-point and Spieker
conics discussed here can still be made. In fact, it
is quite likely that using dynamic geometry
software in teaching geometry at high school or
tertiary level may enable learners and students to
more easily make their own discoveries, as the
author has found on several occasions when
working with prospective and in-service
mathematics teachers. In particular, dynamic
geometry software encourages an experimental
approach that enables students to make and test
geometric conjectures very efficiently.
In recent years there has been a general increase
in geometry research on many fronts. We’ve seen
the development and expansion of Knot Theory
and its increased application to biology, the use of
Projective Geometry in the design of virtual reality
programs, the application of Coding Theory to the
design of CD players, an investigation of the
geometry involved in robotics, use of Search
Theory in locating oil or mineral deposits, the
application of geometry to voting systems, the
application of String Theory to the origin, nature
and shape of the cosmos, etc. Even Soap Bubble
Geometry is receiving new attention as illustrated
by the special session given to it at the Burlington
MathsFest in 1995.
Even Euclidean geometry is experiencing an
exciting revival, in no small part due to the recent
development of dynamic geometry software such
as Cabri, Sketchpad and Cinderella. Indeed, Philip
Davies (1995) already ten years ago predicted a
possibly rosy, new future for research in triangle
geometry. Just a brief perusal of some recent issues
of mathematical journals like the Mathematical
Intelligencer, American Mathematical Monthly,
The Mathematical Gazette, Mathematics
Magazine, Mathematics & Informatics Quarterly,
Forum Geometricorum, etc. easily testify to the
greatly increased activity and interest in traditional
Euclidean geometry involving triangles,
quadrilaterals and circles. Of note too is a specific
Yahoo discussion group which is specifically
dedicated to current research in triangle geometry
and traditional Euclidean geometry. Readers are
invited to visit:
http://groups.yahoo.com/group/Hyacinthos/
It is therefore unfortunate that, with the
exception of a handful of South African
universities, hardly any courses are offered in
advanced Euclidean, affine, projective or other
geometries. In this respect, we seem to be lagging
behind some leading overseas universities where
there is a resurgence of interest in geometry not
only at the undergraduate, but also at the
postgraduate and research level. Not only does this
tendency in South Africa narrow the potential field
of research for a young mathematical researcher,
but especially impacts negatively on the training of
future mathematics teachers, who then return to
teach matric geometry, having studied no further
than matric geometry themselves.
In contrast to the present South African tertiary
scene, the Mathematics Department at Cornell
University, for example, is currently running more
geometry courses at the graduate level (our
postgraduate level) than any other courses
(according to a personal communication to the
author from David Henderson about four or five
years ago). Moreover, Peter Hilton, one of the
leading algebraic topologists (now retired from
Binghamton University), is well known for
frequently publicly stating that geometry is a
marvelous and indispensable source of challenging
problems, though algebra is often needed to solve
them. It is also significant that the recent proof of
Fermat's Last Theorem by Wiles relied heavily on
many diverse fields in mathematics, including
fundamental geometric ideas (see Singh, 1997).
Note: A Dynamic Geometry (Sketchpad 4) sketch
in zipped format (Winzip) of the results discussed
here can be downloaded directly from:
http://mysite.mweb.co.za/residents/profmd/spieker.zip
(This sketch can also be viewed with a free
demo version of Sketchpad 4 that can be
downloaded from:
http://www.keypress.com/sketchpad/sketchdemo.html)
References
Anthony, J.M. (1994). In Eve's Circles. Notes of
the Mathematical Association of America, 34.
Washington: MAA.
Coolidge, J.L. (1971). A Treatise on the Circle and
the Sphere (pp 53-57). Bronx, NY: Chelsea
Publishing Company (original 1916).
Honsberger, R. (1995). Episodes in Nineteenth &
Twentieth Century Euclidean Geometry (pp 7-
13). The Mathematical Association of America.
Washington: MAA.
Davies, P.J. (1995). The rise, fall, and possible
transfiguration of triangle geometry. American
Mathematical Monthly, 102(3), 204-214.
De Villiers, M. (1996). Some Adventures in
Euclidean Geometry, University of Durban-
Westville (now University of KwaZulu-Natal).
Michael de Villiers
37
De Villiers, M. (1997). The future of secondary
school geometry. Pythagoras, 44, 37-54. (A pdf
copy can be downloaded from:
http://mysite.mweb.co.za/residents/profmd/futur
e.pdf )
De Villiers, M. (2000). Generalizing Van Aubel
using duality. Mathematics Magazine, 73(4),
303-306. (A pdf copy can be downloaded from:
http://mysite.mweb.co.za/residents/profmd/aube
l.pdf )
De Villiers, M. (2005). A generalisation of the
nine-point circle and Euler line. Pythagoras, 62,
31-35.
Gutierrez, A. Triangle Centers at:
http://agutie.homestead.com/files/Trianglecenter_B
.htm
Kimberling, C. Triangle Centers at:
http://faculty.evansville.edu/ck6/tcenters/index.htm
l
Kimberling, C. Encyclopedia of Triangle Centers
at:
http://faculty.evansville.edu/ck6/encyclopedia/
Pyenson, L. (1985). The Young Einstein: The
Advent of Relativity. Boston: Adam Hilger.
Singh, S. (1997). Fermat's Last Theorem. London:
Fourth Estate Publishers.
At the age of 12, I experienced a second wonder of a totally
different nature: in a little book dealing with Euclidean plane
geometry, which came into my hands at the beginning of the
school year. Here were assertions
, as for example the
intersection of the three altitudes of a triangle in one point,
which -- though by no means evident --
could nevertheless be
proved with such certainty that any doubt appeared to be out
of the question. This lucidity and certainty made
an
indescribable impression on me.
– Albert Einstein (Autobiographical Notes)
