PUBLIKACIJE ELEKTROTEHNICKOG FAKUL TETA UNIVERZITETA U BEOGRADU
PUBLICATIONSDE LA FACULTED'ELECTROTECHNIQUEDE L'UNIVERSITE A. BELGRADE
SERlJA:
MATEMA
TlKA
I F I Z I K A
ON THE RATIO
SERlE:
MA THEMA
-
M! 460 (1973)
mambme/abc
FOR THE
N!! 412
452.
-
TlQUES
TRIANGLE
ET PH Y S I QUE
ABC*
Gene Bottema and Mirko Jovanovic
1. The image plane. For the triangle ABC with sides a, b, c and medians
ma' mb, me we investigate the ratio p=mambme/abc.
We have 4m/=-a2+2b2+2c2
a.s.o; therefore
(1.1)
or if,
(1. 2)
a2=x,
( 1.3)
(-x+2y+2z)
In
represent
point p
(Fig. 1).
As
b2=y,
(2x-y+2z)
c2 =Z,
(2x+2y-z)-AXYZ=0.
an image plane we consider the equilateral triangle XYZ and we
the triangle ABC (more precisely: the class of similar triangles) by the
the distances of which to YZ, ZX, XY are proportional to x, y, z
a consequence of the triangle inequalities for a, b, c we have
(1.4)
hence a point p is an image point if and only if it is inside the inscribed
circle C of Xyz. Points on C represent degenerated triangles. Let Xl, Y1, ZI'
be the midpoints of YZ, ZX, XY. If CI.= nl2 we have x- y-z = 0, which is
the equation of the line YtZl' Hence the image points of rectangled triangles
are on the line segments Y1Zl' ZI Xl' Xl Y1; those of acute triangles are inside
Xl Y1Zl; the image points of obtuse triangles are inside the three segments of C
outside Xl Y1ZI' The images of isosceles triangles are on the line segments
Xl X2' Y1Y2, ZI Z2' X2 being the second intersection of C and. XXI a. s. o.
2. The pencil of cubic curves. The image plane introduced here may be of use
for the study of relationships which are rational in x, y, z as is the case
for (1.3). This equation represents for a fixed value of A a cubic curve k
with the same symmetry as the equilateral triangle; for variable A it is that
of a pencil of such curves k.
*
Presented
July 27, 1973 by D. S. MITRINOVH';and R. R. JANIe.
197
o. Bottema and M. Jovanovic
198
-x + 2y + 2z = 0 is the equation
of the line mJ through
tangent to C, intersecting XY and ZX at
divide YZ in three equal parts.
831
and
S21 (Fig.
X2 parallel
2), a.s.o.;
to YZ,
SJ3 and
The curve k for A= 0 is degenerated into the lines mi; for A=
S12
00
it
consists of the sides of Xyz. Hence the nine base points of the pencil (1.3) are
the six points Sij and the points SJ' S2' S3 of YZ, ZX, XY at infinity.
It follows from this that the pencil contains another curve degenerated
into three lines, viz. the line 11 through S1' S23' S32 and two similar lines 12,/3;
they are the lines through the center M of XYZ, parallel to the sides. The
equation of 11 reads - 2 x + y + z = 0; for the curve k consisting of
11' 12, 13we
obtain A= 27. Therefore the triangles ABC for which p = 2- V3 (that is the
8
value for the equilateral triangle) are those for which 2a2 = b2 + c2, and cyc!.
They are either acute, or rectangled (with sides proportional to V3, V2, vI),
or obtuse.
The pencil contains one more degenerated curve: that consisting of the
line at infinity and the circle through the six points Sij' As it lies outside C
its points are not images of real triangles; it corresponds to A= -27.
3. Isosceles triangles. We determine first the isosceles triangles ABC (with vertex
A, say) for which p has a given value. From b = c it follows that x: y : z = [L: 1 : 1
and (1.3) gives us
(3.1)
For the discriminant
of this cubic equation we obtain after some algebra
(3.2)
We know that A= 27 corresponds to the equilateral triangle; (3.1) has
three equd roots [L= 1 in this case. For all other (positive) values of A we
have D>O, which implies that (3.1) has one real root [La' easily seen to satisfy
the inequality
(3.3) .
while [Lais decreasing if A increases from zero to infinity.
For a rectangled triangle we have [La= 2 and therefore A= 25. The conclusion is: there is always one isosceles triangle for any value of p; it is obtuse
.
.
5
.
5
5
I f P < -,
acute If p > - .
rectang 1e d If p = -,
8
8
8
4. The cubic curve. If k has a double point it has in view of its symmetry
either at least three double points or it passes through M or through the
isotropic points of the plane. In all these Cases it is degenerated. Hence if k is
not degenerated it is a curve of genus one.
As k passes through Si it has three asymptotes qi parallel to the sides
of Xyz. For q1 we obtain
(4.1)
9 (-x+
2y+ 2Z)-AX=O;
On the ratio ma mb mc/abc
for the triangle
ABC
199
its three intersections with k coincide at SI; hence S; are the three (real) inflexion points of k. For A= 0 ql coincides with ml, for A= 00 with YZ, for A= 27
with II' It intersects XXI at TI =(36, A+9, A+9); for A<27 TI is between X2
and M, for A> 27 it is between M and XI, As the unique intersection of k and
XXI follows from section 3 we are able to draw a sketch of k for A>O. If A<27
we obtain Fig. 3, for A>27 the situation is given in Fig. 4. Hence for any A k
has points inside the circle C; only if A> 25 there are p,ints inside XI YI ZI'
but in this case k contains points outside XI YI Zl as well. The conclusion IS
for obtuse triangles: O<p< +
for rectangled triangles:
(4.2)
for acute triangles:
00,
~ -;;;'p<
8
5
8
-<p<
+
+
00
,
00.
These inequalities cannot be improved.
m,
x
/,
S,
y
z
Fig. 1
Fig. 2
y
Fig. 3
z
Fig. 4