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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)