Triangles Having a Common Meanv

On Triangles Having a Common MeanAuthor(s): O. J. RamlerReviewed work(s):Source: The American Mathematical Monthly, Vol. 47, No. 3 (Mar., 1940), pp. 140-145Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2304214 .

Accessed: 05/01/2013 01:43

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access toThe American Mathematical Monthly.

http://www.jstor.org

This content downloaded on Sat, 5 Jan 2013 01:43:54 AMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=maa

http://www.jstor.org/stable/2304214?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


140 ON TRIANGLES HAVING A COMMON MEAN [March,

ON TRIANGLES HAVING A COMMON MEAN*

0. J. RAMLER, The Catholic University of America

Introduction. The term "mean" is used here in the sense defined by P. De- lens [1]. C. E. Van Horn [2] has considered the same triangle and called it "the equilateral derivative" of a triangle. Van Horn gives the following construction for the mean triangle, or the equilateral derivative of a given triangle: Let ABC be any triangle and let 0 be its circumcenter. Let D, E, F, be the midpoints of the sides BC, CA, AB, respectively. Draw the side bisector DH of the side BC to cut the arc BA C of the circumcircle containing the opposite vertex A at the point H. The side bisectors EJ, FK are similarly drawn to cut the arcs ABC and BCA at the points J and K, respectively. Choose the point L on arc HA so that HL is one-third of the arc HA. Select the points M and N in a similar manner on the arcs JB and KC, respectively.

Employing conjugate coordinates we take the circumcircle of the fundamental triangle AiA2A3 to be the unit or base circle and let the co6rdinates of the vertices A, be the turns ai, (i= 1, 2, 3). We consider ao to be roots of the equation t3 - o1t2 +?2t- 03=0. Then as Delens shows, the vertices of the mean triangle of A1A2A3 are 0-31/3, 0y31/i3, 0.31/3 2, where 1 + +X2 = 0. It is the purpose of this paper to discuss the relation of two triangles having a common mean triangle, and to apply some of the results to theorems discussed by Musselman in his article in this MONTHLY, On the line of images [3].

1. Mutually orthopolar triangles. Let si be the elementary symmetry functions of the vectors /3i of the vertices Bi of a second triangle inscribed in the base circle. From the definition it follows at once that the two triangles Ai and Bi have a common mean when 03 = S3. The vectors to the orthocenters of triangles Ai and Bi are oi and si, respectively. The midpoint m of the segment joining the orthocenters is given by

yi + Si Oi (l . l) 1M+=

- - + - (01 + 02 + 0_30i10:21) 2 2 2

when the triangles have a common mean. The right member of equation (1.1) identifies m as the orthopole of side B,B2 as to triangle A . The symmetry of the expression (o;?+sl)/2, however, leads to results given by Murnaghan [4] and Godeau [5 1:

THEOREM I. If two triangles inscribed in the same circle have the same mean, the midpoint of the segment joining their orthocenters is the orthopole of any side of one triangle with respect to the other. In other words, the two triangles are mutually orthopolar.

We have at once the following: * Presented at the meeting of the Maryland-District of Columbia-Virginia Section of the

Association at Washington, December 9, 1939.



1940] ON TRIANGLES HAVING A COMMON MEAN 141

COROLLARY. The center of the nine-point circle of a triangle is the orthopole of any side of its mean.

2. The line of images. If we take any point T on the circumcircle of a triangle A1A2A3 and reflect this point in the sides of the triangle we obtain three points lying on a line, the line of images of the point T. This follows at once from the property of the pedal line of T as to the triangle A1A2A3, and since the pedal line bisects the segment joining the orthocenter to the pole T, it follows that the line of images passes through the orthocenter. Moreover, the line of images is the directrix of the parabola having T for focus and inscribed in the triangle A1A2A3. Musselman [3] has given the equation of the line of images of T as to A1A2A3 to be

(2.1) Tx-03-t= To1-, .

Now consider a second triangle B1B2B3 inscribed in the base circle, and let /3 be its vertices. Then si is its orthocenter and the equation of the line of images of the same point T as to this triangle is

(2.2) Tx -S33C = Ts -s2.

Lines (2.1) and (2.2) are coincident when 03=s3 and To1-02=Ts1-s2, i.e., T=(0f2-s2)/(0f1-s1). Hence the following:

THEOREM II. If two triangles inscribed in the same circle have a common line of images for the same point on their circumcircle, they have a common mean and a common inscribed parabola whose focus is the given point.

Theorems I and II are equivalent to a theorem stated by Cwojdzinsky [6]: When two triangles are inscribed in a circle and circumscribed to a parabola, the midpoint of the distance of their orthocenters is the orthopole of any side of one of the triangles for the other.

The result stated in Theorem II may be verified otherwise. It is well known that if, from a point t of the circumcircle, lines are drawn

to the three sides of an inscribed triangle making equal angles 0 with those sides, the feet of these lines lie on a line which may be considered a generalized pedal line of t under the chosen angle 0. We shall call it a skew pedal line. As the angle 0 varies the pedal line envelopes a parabola inscribed in the fundamental triangle and having t as its focus. Letting e2i0 = l/k we obtain the equation of the skew pedal line of t under angle 0 with respect to triangle A1A2A3 to be (2.3) (k-1)tx + (+ --02-)+ k

k kt

As k varies we obtain the map equation of the envelope to be

(t-a1l)(t-ae2)(t-ae3) (2.4) x = t- -

t2(k - 1)2




which identifies the envelope to be a parabola. If k =ai/t, the equation (2.3) of the corresponding pedal line becomes

x + aoak jCX - j - a?k = O, (i = j = k = 1, 2, 3),

showing that the parabola is inscribed in the triangle A1A2A3. Its vertex tangent is the pedal line for k =-1, namely

(2.5) 2tx-23C + 03t-' + -2-t2 _-1t = O.

If k=/3i/t, where t= (02-S2)/(01-si), and 03=s3, equation (2.3) becomes X+Ij3kX -0j -Ok =0 which is the equation of side B1Bk of triangle B1B2B3. The triangles Ai and Bi have a common mean by virtue of the assumption o-3 = s3. Theorem II is thus verified by direct substitutions. From the results obtained above, we are led to the observation that if two triangles inscribed in the same circle have a common line of images for the same point of their circumcircle they are mutually orthopolar, which also implies that they have a common mean.

The line of images common to two triangles having the same mean is also the locus of points R mentioned in Musselman's generalization of Canon's theorem [7]. The generalized Canon theorem may be stated as follows: If, for every point R on the line HP, where H is the orthocenter of A1A2A3 and P is any point in the plane, we determine the images C1C2C3 of R in the sides A2A3, A3A1, A1A2, the four circles C1 C2A3, C2C3A1, C3CIA2, and A 1A2A3 meet in a fixed point M. Now if Ai and Bi are two triangles having the same mean, and if we choose P to be the orthocenter s, of B,, the point M has the coordinate (0f2-s2)/(0-1 -sI). The symmetry of this result enables us to state the following modification of the generalization of Canon's theorem:

THEOREM III. If Ha and Hb are the orthocenters of two triangles A1A2A3 and B1B2B3 having the same mean, and if for any point R on HaHb we determine the images A,' and B,' in the sides of triangles Ai and B, respectively, the circles A1'A2'A3, A2'A3'A1, A 3'A1'A2, B1'B2'B3, B2' B?' B1, B 'B'B2 meet on the base circle at a point which is the focus of the parabola inscribed in the two triangles A and B,.

It can be readily shown, too, that A, and B,' are images in the common line of images HaHb.

3. Co-mean triangles and a theorem of Blanc. If any transversal be drawn through 0, the circumcenter of A1A2A3, cutting the sides A2A3, A3A1, A1A2 in Cl, C2, C3, respectively, the three circles with AiCi as diameters meet in two points, one on the circumcircle and the other on the nine-point circle of AjA2A2. Their common chord passes through H, the orthocenter of triangle Ai [8]. If the transversal is taken as the diameter through T on the circumcircle, Mussel- man [7] has shown that the three circles A,C, meet at

0-3 ? oXT2 1 0f3\ xc = ~ an d- XN=- j T--

0-2+ T 2 ~2 \ T 2



1940] ON TRIANGLES HAVING A COMMON MEAN 143

on the circumcircle and nine-point circle respectively, and that the equation of the common chord of the three circles is

(3.1) T2(o2 + T2)X - 03(03 + LTT2)x +023 - 01T.4 =

Again we consider a second triangle B1B2B3 having a common mean with triangle A1A2A3, and we find that we can identify line (3.1) with the common line of images of point (0f2-S2)/(0i--Sl), providing

(3.2) T 2(o-2 + T2) 0:3 + o-1T2 o-1T4 -0-20f3 02 - S2 01 - S1 02S1 - of1S2

These equations are consistent when T is a root of

(3.3) (f1 - si)T4 - (0-2S1 - o1s2)T2 -0-3(0-2 - S2) = 0

Now the equation of the line of images of point (0f2-S2)/(01-si) as to triangles A and B is

(3.4) X(0f2 - S2) - 0-3X(0l - Si) - S10-2 + 0-1S2 = 0.

This intersects the circumcircle xl = 1 in two points xl, X2, roots of the quadratic

(3.5) x2(02 - S2) - (S1(J2 - 01S2) X- 0-3 (0-1 - Si) = 0 .

Let x --3/T2; then equation (3.5) becomes identical with equation (3.3), enabling us to devise a means to construct the four points which represent the roots of equation (3.3). The equation x = -3/T2 identifies the points x, T, and - T as vertices of a right triangle having the same mean as the fundamental triangle A1A2A3. There are therefore two distinct diametral transversals OT which yield the same common chord of the system of circles mentioned in Blanc's theorem when applied to two triangles having a common mean. We have, then, the following construction for the points T: If L is a vertex of the mean triangle, and X1 is an intersection of the common line of images with the circumcircle, locate a point K on the circumcircle so that arc LX1=2 arc KL and point L lies between X1 and K. Then one of the diametral transversals will be the diameter perpendicular to OK. The other diameter is found by using the second intersection X2 of the common line of images with the base circle.

From equation (3.2) we have T2 = Xc a, where a = (0f2-S2)/(of1-Si), the point whose line of images is (3.4). Then we may write, since there is a point Xc corresponding to a point T2, X 'a = T12 and Xc" a= T22. From these relations we find

/1 - '~"2 2~' 02S1 - 0-1S2 X' + X = (T 2+ T2?)/a = X= -+X2, 0-2 - S2

Xc'X" = T T22/a2 = - 0-3(0-1 -Si) XiX2 0-2 -S2

where x1 and X2 are the roots of the equation (3.5). These results involve only a point on the circumcircle and its common line of images as to two triangles




having a common mean, and since, as we have seen, the point and line are focus and directrix, respectively, of a parabola common to the two triangles, we may state the following:

THEOREM IV. All triangles in a poristic system of triangles inscribed in a circle and circumscribed to a parabola have a common mean. Each of the two diameters of the circle bisecting the arcs joining the focus to the points where the directrix cuts the circle, cuts the sides of any triangle of the poristic system in points such that the circles on the segments joining these points to the opposite vertices as diameters meet at a point common to the circle and the directrix.

4. Isogonal conjugates. THEOREM. The isogonal conjugates of the common orthopole of two mutually orthopolar triangles in those triangles are symmetric with respect to the circumcenter.

Let the mutually orthopolar triangles be A1A2A3 and B1B2B3. Let ai and /3 be their vertices, respectively. Then if P(p) is their common orthopole, 2P =o?+si and =S3. If P does not lie on the circumcircle, its isogonal conjugate x is given by P+x+o3 f=o [9] for the triangle A1A2A3. Similarly, the isogonal conjugate y of p in triangle B1B2B3 is given by P+y+o8039 =s1. Adding we get, remembering that 2p= o+sl, x+y+03p(x+y) =0, which is not true un- less x+y=O, i.e., x= -y, and the points are symmetric with respect to the circumcenter.

5. Perspective triangles. Suppose triangles A 1A2A3 and B1B2B3 are mutually orthopolar and perspective from a point P(p). Then

ai-p (al- P)(a2 - P)(ax3- p) i = , (i = 1, 2, 3), and I1I2I83 = 3= - )(a3-i

aip (alp 1Q2 - 1)(a2fi- 1

hence, regarding triangle A1A2A3 as fixed, and B1B2B3 as variable,

p3 - 01p2 + 2p + i C3p- 2fi2 + 1fP=23 - 2?0_2P ? o0.3 0-020-3P ? a1o3 f= 20-3

is the equation of the locus of the centers of perspective. The locus is a cubic cutting the base circle zs= 1 at the vertices of the fundamental triangle A1A2A3, and at the vertices of their common mean. The cubic cuts the sides A jAk

where pi =ai(30-3-aioJ2)/(0J3- a,3), i.e., where the Lemoine axis crosses the sides of A1A2A3. The clinant at any point is given by

dz 3of22 - 20-20y3z + 0-10-3

dt 3Z2 - 2of1z +?02

which shows that the cubic cuts McCay's cubic [10] orthogonally [11]. The asymptotes are z = - COi32I3z+13(0-1+C0ir20-3 1/3), where cow is any one of the cube roots of unity. The asymptotes intersect at the centroid of A1A2A3, Z =0-i/3.

It may be remarked here that if the pair of perspective triangles A1A2A3 and B1B2B3 have opposite means in the sense defined by Delens [1], i.e., if a1a2a3 = -0313203, the locus of the centers of perspective is McCay's cubic [11].



1940] ON EXTREMA OF FUNCTIONS 145

References 1. P. Delens, Mathesis, vol. 51, 1937, p. 264. 2. C. E. Van Horn, this MONTHLY, vol. 45, 1938, p. 435. 3. J. R. Musselman, this MONTHLY, vol. 45, 1938, p. 421. 4. F. D. Murnaghan, Mathesis, vol. 41, 1927, p. 27. 5. R. Godeau, Mathesis, vol. 41, 1927, p. 72. 6. Cwojdzinsky, Archiv der Mathematik und Physik, 1902, p. 316. 7. J. R. Musselman, this MONTHLY, vol. 45, 1938, p. 424. 8. C. Blanc, Nouvelles Annales de Math6matique, Third Series, vol. 19, 1900, p. 573. 9. F. V. and F. Morley, Inversive Geometry, p. 196. 10. W. S. McCay, Transactions of the Royal Irish Academy, vol. 29, 1889, p. 313. 11. J. H. Weaver, this MCNTHLY, vol. 42, 1935, p. 497.

ON EXTREMA OF FUNCTIONS WHICH SATISFY CERTAIN SYMMETRY CONDITIONS

R. F. RINEHART, Case School of Applied Science

1. Introduction. The following theorems on maxima and minima are well known.

I. The function Xl X2 . . . Xn, where the real variables xi, X2, x, x are subject to the condition xl+x2+ +x, =c>O, has a proper relative maximum atX1=X2= ... = x=c/n.

II. The function Xl+X2+ ?+xn, where the real variables xi, x2, * ,

are subject to the condition XlX2 x.. x,=c>O, has a proper relative minimum at Xl =X2 = *.** = Xn = -c.

R. H. Garver* has pointed out that most of the elementary problems on the applications of the theory of maxima and minima which are customarily en- countered in textbooks on the calculus, can, by appropriate transformation of the variables, be put into forms to which Theorems I or II may be applied. One infers that Theorems I and II are rather fundamental results in the theory of maxima and minima.

Theorems I and II are, however, susceptible of a sweeping generalization. It is the purpose of this paper to call attention to this generalization. t

A mild study of Theorems I and II leads one rather naturally to suspect that the essence of those theorems may lie, not in the use of the particular functions x1+x2+ + ?x- and XlX2 xn, but in the symmetry of the conditional relation and the function to be extremized, in the variables xl, X2, , xn. This suspicion may be strengthened by the construction of examples which can not be made to fall under the jurisdiction of Theorems I or II by any transformation of the variables.

* This MONTHLY, vol. 42, 1935, pp. 435-437. t I have been informed by Professor Tibor Rado that some form of this generalization is

known to Fejer, who remarked about it several years ago in the course of a conversation with him. However, the result does not seem to be generally known, and I have not been able to find it in the literature.



Triangles Having a Common Meanv

Documents

Transcript of Triangles Having a Common Meanv