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1920.] ON THE ORTHOCENTRIC QUADRILATERAL. 199 

value of the function when x = 1, that is, the sum of the coefficients of the func- 
tion. As the arms are turned from 0° to 90°, keeping the slides properly adjusted 
and all the cords taut, the distance of this slide, R, from its initial position will 
be continuously the value of the function as x varies continuously from 1 to 0. 



ON THE ORTHOCENTRIC QUADRILATERAL. 1 

By NATHAN ALTSHILLER-COURT, University of Oklahoma. 

Introduction, (a) The altitudes AD, BE, CF of a triangle ABC meets in a 
point H, the orthocenter of ABC. The triangle DEF formed by the feet D, E, F, 
of the altitudes is frequently called the orthic triangle of ABC. 

To Carnot 2 is due the credit for having called attention to the almost obvious 
fact that each of the four points H, A, B, C, is the orthocenter of the triangle formed 
by the other three. 

The points A, B, C, H are referred to as an orthocentric group of points, or an 
orthocentric quadrilateral, and the four triangles determined by these four points 
as an orthocentric group of triangles. 

(b) In 1821 Brianchon and Poncelet showed that the circu'mcircle (2V) of 
the orthic triangle DEF of ABC passes through the mid-points A', B', C, of the 
sides BC, CA, AB, of ABC, and also through the mid-points P, Q, R, of the 
segments AH, BH, CH respectively. 3 That the circle through the first six 
points mentioned passes also through the last three becomes obvious if we observe 
that DEF is the orthic triangle not only of ABC, but of each of the four triangles of 
the orthocentric group ABCH. 

(c) In 1822 Feuerbach proved 4 that the circle (2V) is tangent to the four circles 
which touch the sides of the triangle ABC. It was not until 1861 that Sir William 
R. Hamilton pointed out that (N) is also tangent to the circles touching the sides 
of the triangles BCH, CHA, HAB? Now since the orthic triangle DEF is com- 
mon to the four triangles of the orthocentric group ABCH, the circumcircle (N) 
of DEF is the nine-point circle of each of these four triangles, and therefore 
Hamilton's extension of Feuerbach's theorem becomes self-evident. 

1 Read before the American Mathematical Society, St. Louis, December 31, 1919. Readers 
of this article will be interested in comparing it with the first part of the author's earlier article 
" On the I-centre of a triangle " (1918, 241-246)— Editor. 

2 Carnot, De la correlation des figures de Geometrie, 1801, p. 102. 

3 For the proof, see, for instance, J. Casey, A Sequel to Euclid, second edition, 1882, p. 58, or 
C. V. Durell, Plane Geometry for Advanced Students, vol. 1, pp. 30-31. 

4 For a proof see Casey, I.e., pp. 58-61, or Durell, I.e., pp. 46-47 and pp. 149-150. 

5 In making this statement Professor Altshiller-Court was possibly misled by Casey's refer- 
ence to the result as "Sir William Hamilton's Theorem" (Quarterly Journal of Mathematics, 1861, 
p. 249) and by the fact that Sir William proposed the result as a problem in Nouvelles annates de 
mathematiques, 1861, vol. 20, p. 216. 

The result was not, however, given originally", by Sir William, but by T. T. Wilkinson, as 
prize-problem 1883 in Lady's and Gentleman's Diary, London, 1854, p. 72 (Solutions, Diary, 
1855, pp. 67-69).— Editor. 



200 



ON THE ORTHOCENTRIC QUADRILATERAL. 



[May, 



These examples suggest that in certain connections it may be fruitful to con- 
sider the circle (2V) as belonging not to the triangle ABC, but to the orthocentric 
group ABCH. The following considerations are based on this remark. 

1. The center N of the nine-point circle (N) of the triangle ABC was 
shown by Benjamin Bevan, in 1804, 1 to lie midway between the orthocenter H 
and the circumcenter of the triangle ABC. In other words, the circumcenter 
of ABC is the symmetric of H with respect to N. But, as has been pointed 
out above, (N) is also the nine-point circle of the triangle BCH, whose orthocenter 
is the point A, hence the circumcenter 0i of BCH is the symmetric of A with 




respect to N. Similarly for the circumcenters 2 , 03, of the triangles CHA, 
HAB. Consequently: The four circumcenters of an orthocentric group of triangles 
form an orthocentric group which is the symmetric of the given group with respect to 
the nine-point center. 

2. From the symmetry of the two groups of points ABCH and O1O2O3O 
follow immediately all the known properties of the circumcenters 0\, 02, 03, 0. 
For instance: 

(a) The triangles 010203 and ABC are congruent 2 and furthermore, their sides 
are respectively parallel. It may also be observed that these properties hold for 
the pairs of triangles 2 3 and BCH; 3 00i and CHA; 0Ox0 2 and HAB. 

1 For the proof compare Casey, or Durell, I.e. 

2 Durell, I.e., p. 36, exercise 89. 



1920.] ON THE ORTHOCENTKIC QUADRILATERAL. 201 

(b) The point H is the circumcenter of the triangle O1O2O3. 1 Similarly the 
points A, B, C, are the respective circumcenters of the triangles O2O3O, O3OO1, 
OO1O2. 

(c) The lines BOz and CO% are parallel and similarly for other pairs of analogous 
lines. 

3. A wealth of other propositions, so far unobserved or unannounced, may be 
derived from the two symmetrical figures. We shall call attention to the 
following. In the symmetrical transformation considered the center of sym- 
metry N is a double point, and the nine-point circle (N) is transformed into itself, 
hence : An orthocentric group of triangles and the orthocentric group of their circum- 
centers have the same nine-point circle. 

4. The orthic triangle D'E'F' of the orthocentric group OO1O2O3 is the 
symmetric of the orthic triangle DEF of the orthocentric group ABCH, the pairs 
of points D, D'; E, E'; F, F'; being diametrically opposite on the circle (N). 
Thus we find a geometric interpretation of three new points of the nine-point circle 
of the triangle ABC. 

5. The nine-point circle (N) of the orthocentric group OO1O2O3 is tangent 
to the sixteen circles which touch the sides of the four triangles of this group, 
according to Hamilton's extension of Feuerbach's theorem (Introduction). 
These sixteen circles are the symmetric, with respect to N, of the analogous 
sixteen circles of the orthocentric group ABCH. Thus we find sixteen new circles 
which are tangent to the nine-point circle of the triangle ABC. 2 

6. The orthocentric group OO1O2O3 has been derived by symmetry from the 
given orthocentric group ABCH. But the process may be reversed, and the 
group HABC may be derived from the group OO1O2O3 considered as given. 
Consequently: The vertices of a given orthocentric group of triangles may be con- 
sidered as the four circumcenters of a second orthocentric group of triangles, the two 
orthocentric groups having the same nine-point circle and being symmetrical with 
respect to its center. 

7. The point of intersection G of the medians of ABC, often referred to as the 
centroid of ABC, lies, according to a theorem of Euler, 3 on the line joining the 
orthocenter H to the circumcenter of ABC, and we have GO/GH = 1/2. 
Since the nine-point center N is the midpoint of OH, we have NGjNH = 1/3, 
the points G and H being on opposite sides of N. In other words, the point G 

1 Durell, I.e., p. 36, exercise 88. 

2 The results of paragraphs 1, 2, 3, 4 and 5 were given by T. T. Wilkinson in Mathematical 
Questions with their Solutions from the Educational Times, London, Vol. 1, 1864, pp. 6-7; see also 
Mathematical Questions, etc., Vol. 6, 1866, pp. 25-26. 

T. T. Wilkinson seems to have been the first to discover an infinite series of circles tangent to 
the nine-point circle of a triangle (Lady's and Gentleman's Diary, London, 1857, p. 86; 1858, 87) : 
"// the radical centers of the inscribed and escribed ■circles of any triangle be taken, and circles be 
inscribed and escribed to the triangles formed by joining these radical centers, and the radical centers 
of the latter system of circles be again taken and circles inscribed and escribed to the triangles thus 
formed, and so on ad infinitum, the infinite number of circles thus found, as well as the original system 
of inscribed and escribed circles, always touch the circle drawn through the middle -points of the [sides 
of the] first triangle." — Editor. 

3 Durell, I.e., p. 41. 



202 17 20 [May, 

corresponds to H in a similitude of ratio — 1/3, the center of similitude being N. 1 
But N is also the nine-point center of the triangle BCH, whose orthocenter is A, 
hence the centroid Gi of BCH corresponds to A in a similitude of ratio — 1/3 
with N as center of similitude. Similarly for the centroids 6? 2 , 63, of the triangles 
CHA, HAB. Consequently: The four centroids of an orthoeentric group of tri- 
angles form an orthoeentric group, the two groups being similar and similarly placed. 

8. Since the centroids G, G\, G 2 , G3, form an orthoeentric group, all the 
properties of such a group immediately follow, as, for instance, that 6? is the 
orthocenter of the triangle G1G2G3, etc. 

Again the similitude of the two groups GG1G2G3 and HABC puts into evidence 
a great many properties, as for instance, that G1G2 is parallel to AB and is equal 
to 1/3 of its length; that the point of intersection of GGi and G2G3, which will 
be represented by (GG\, G 2 G 3 ), is collinear with N and D = (HA, BC) ; etc. 
The reader may find it interesting to formulate a number of these propositions. 

9. In the similitude (7) by which the group GG1G2G3 is derived from the group 
HABC, the center of similitude N is a double point. Hence: An orthoeentric 
group of triangles and the orthoeentric group of their centroids have the same nine-point 
center. 

10. The orthoeentric group GG1G2G3 has been derived from the given ortho- 
centric group HABC by a similitude of center N and ratio — 1/3. But the 
process may be reversed, and the orthoeentric group HABC may be derived from 
the orthoeentric group GG1G2G3, considered as given, by a similitude of ratio — 3, 
the center remaining the same. Consequently : The four points of an orthoeentric 
group may be considered as the centroids of another orthoeentric group of triangles, 
the two groups having the same nine-point center, this point being a center of similitude 
of the two groups, the ratio of similitude being — 3. 

11. Since from (1) the two groups HABC and OO1O2O3 are symmetrical 
about the center N, therefore it follows from (10) that the two groups GG1G2G3 
and OO1O2O3 admit N as a center of similitude, the ratio of similitude being + 3. 
Hence: The centroids and the circumcenters of an orthoeentric group of triangles 
form two orthoeentric groups of points having the same nine-point center, this point 
being a center of similitude of these two groups, the ratio of similitude being + 3. 



1720 

C. Maclaurin's Geometria organica sive descriptio linearum curearum universalis, 
published at London — G. Poleni's De mathesis in rebus physicis utilitate praelectio 
habita • • •, published at Patavia — Second edition of L'Hospital's TraitAanalytiques 
des sections coniques, published at Paris — Alexandre Sav6rien, author of Diction- 
naire universel de matMmatiques et de physique (2 vols., Paris, 1753), born July 16. 

1 Euler, Novi comment, acad. sc. Petrop., vol. 11 (1765), 1767, p. 114. — Editor.