Problems & Puzzles:
Problems
Problem 64. Prime convex
quadrilateral
Dmitry Kamenetsky sent the following nice puzzle:
Can you find a convex
quadrilateral such that all four of its sides and two
diagonals have lengths such that:
Q1.
The quadrilateral has the smallest area and all four
of its sides, two diagonals and area are distinct
integers?
BTW,
Dimitry has already sent one solutions to Q1 that I
will show next week.
Q2. The
quadrilateral has the smallest area and all four of
its sides, two diagonals and area are distinct
prime
integers?
Contributions came from Jan van Delden, Emmanuel Vantieghem
and Dmitry Kamenetski. BTW, the three of them found the same minimal
area solution for Q1 (see diagram below).
***
Jan wrote:
Q1:
Q2:
The quadrilateral with
diagonals having integer lengths is composed of 4 Heronian triangles
(top/down and left/right).
An Herionian triangle is a triangle where the lengths of the sides
are integer.
It is known that at
least one side has an even length and the lengths 1 and 2 are not
possible.
Therefore there is no Heronian triangle with all sides prime.
Furthermore the area is always divisible by 6, so a prime area is
not feasible***
Emmanuel wrote:
Q1. My 'champion' is the
quadrilateral with sides 10, 17, 28, 35, diagonals 21, 39
and area 378.
(the diagonal with length 21
is the one that meets the sides 10, 35).
Q2. This is impossible.
In my opinion, the simplest
way to prove this is the following :
Suppose ABCD is a
quadrilateral with sides a = AB, b = BC, c = CD, d = DA
and diagonals p = AC, q = BD.
Suppose all these quantities
are prime.
It is clear that the prime 2
is not involved here (an integeral triangle that has one
side equal to 2 has at least one other side even).
Let s = (a+b+c+d)/2 be the
semiperimeter of ABCD. The area k of ABCD then can be
computed by the formula
k = SQRT( (s  a) (s  b) (s 
c) (s  d)  (a c + b d + p q) (a c + b d  p q)/4 ).
It is immediately clear that
the expression under the root cannot be an integer when a,b,c,d,p,
q are odd. Thus, "all quantities q, b, c, d, p, q odd" is
impossible.
***
Dmitry
For Q2, I have searched all
quadrilaterals with sides less than 1500. I have only
managed to find two solutions that give a concave (not
convex) quadrilateral with prime edges. Here is one with the
smallest NON INTEGER area:
A=(0.0,0.0), B=(1051.0,0.0), C=(625.0965746907707,1231.4281433806302),
D=(590.8092293054234,50.925971444162)
then AB=1051, BC=1303,
CD=1181, AD=593, AC=1381 and BD=463,
area=299262.75833496224 (*)
In other news, a guy on the Russian dxdy forum has proved
that a solution for problem 2 is impossible. He proved that
if a quadrilateral has prime sides and diagonals then it
cannot have a prime area.
It can perhaps, have an integer nonprime area....
Hopefully he will send you a translation. If not then I will
translate it for you later.
BTW, Dmitry not only found the
same minimal solution for Q1 than Jan and Emmanuel but he also sent
a integer solution for a pentagon! that will be posted in
another separate problem.
Note (*) Emmanuel Vantieghem has
shown that this solution is impossible for any quadrilateral. See
his comments to Q2 in
Problem 65.
***
