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MATH
INVESTIGATIONS
Consll'uclinU trianules Inom
Ihl'ee Uiuen
pal'ts
Of the 1BO problems, 28 are still looking for a solution!
by George Berzsenyi
T THE RECENT ANNUAL
MAA/AMS meeting in Cincinnati,
I enioyed a wonderful
evening with my mathematician
friends, Stanley Rabinowitz
(the series editor of Indexes to Mathematical
Problemsl, Curtis Cooper
and Robert Kennedy (the coordinating
editor andproblems editor of the
excellent Missouri loutnal of Mathematical
Sciencesl, and Leroy (Roy)
Meyers (who served as the problems
editor of theMathematics Magazine
for several years). Since Roy and I
serve on Stan's editorial board, part
of the conversation was about books
to be published by Stan's company,
MathPro Press, in the near future.
These include theLeningrad Mathematical
Olympiads, 1987-L99 1 i
the Problems and Solutions from
the Mathematical Visitor, 1877-
1896, the NYSMI-AR ML Contests,
1989-1994, and two more volumes
of tt.e Index to Mathematical Pr ob -
7ems, covering the years 1975-1979
and i9B5-1989. All of these should
be of great interest to my readers.
Since both Stan and Roy are
deeply interested in geometry, our
conversation led to some problems
in that area, ard I learned that Roy
was an outstanding expert on the
constructibility of triangles from
given data. More precisely, he found
that there are 185 nonisomorphic
problems resulting from choosing
three pieces of data from the following
list of 18 parts of a triangle:
sides
a, b, c
angles a,9, T
altitudes ho, h.r, h"
medians
ffio,frb,ffi"
anglebisectors to,tb,t"
circumradius R
inradius r
semiperimeter s
(For the sake of brevity, I have omitted
the terms "length of" and "rr.easure
of" in the list above. I will also
assume that the notation is self-explanatory
andf or familiar to all of
my readers.)
My first challenge to my readers
is to reconstruct the 186 problems
mentioned above. As a partial aid, I
wiil retain the numbering given to
the list of problems by Roy; his list
is a variation of one provided earlier
by Alfred Posamentier and William
Wernick in their Advanced Geometric
Constructions (Dale Seymour
Publications, 1988). The interested
reader may wish to consult
chapter 3 of this book for a more
thorough introduction to the topic.
Basically, the problems fall into
four categories:
1. Redundant triples, in which any
two of the three given parts will
determine the third. Of the iB6
problems, only (u, B, yl, (u, $, h"l,
(a, a, Rl fall into this group.
2. Unsolvable problems, which do
not allow for the construction of
a triangle by Euclidean tools (that
is, compass and straightedge).
There are 27 such triples.
3. Solvable problems (by Euclidean
tools). There are 128 such problems.
4. (Jnresolved problerns. These are
listed below, retaining the numbering
given to them by Roy in a
preprint he recently sent me.
I wish to take this opportunity to
thank him for sharing with me and
my readers his wonderful findings.
72. a, mo, to l3L. a, mo, r
8L. hr, fro, tb 135. ho, mo, r
82. ho, frb,to l3B. a, to, t
83. ho, 11761 t6 142. ho, tb, r
84. ho, 1176r t" 143. mo, to, t
85. mo, ft)61 t" 144. mo, t6, r
88. a, t6, t" 149. mr, R, r
89. ct, to, t" I5O. to, R, r
9O. a, to, t" 165. ho, mo, s
ll0. ho, mo, R t72. ho, t6, s
ll7. ha, tb, R
ll8. mo, to, R
ll9. mo, to, R
l2o. ta, tb, R 180. to, R, s
173. mo, t,, s
174. mr, t6, s
179. mo, R, s
To prove the unsolvability of
some of the problems, Roy found
CONTINUED ON PAGE 55
30 JIrY/rttotl$T rss4