Find a triangle with integer sides such that the product of the tangents of its half-angles is (2^8 5^2...
geometry/half.angle.tangents New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
Find a triangle with integer sides such that the product of the tangents
of its half-angles is (2^8 5^2 11^2) / (3^2 7^2 13^2 17^2).
Difficult. No one got it last year.
Find a triangle with integer sides and such that the product of the
tangents of its half-angles is
8 2 2
(2) (5) (11)
--------------------
2 2 2 2
(3) (7) (13) (17) .
-- Jim Buddenhagen <jbuddenh@artsci.wustl.edu>
Show solution
One answer is that the lengths of the sides of the triangle are
6050881, 15629120, 21397761
Anyone want to know how I got it? Just a lucky guess! :-)
Seriously, elementary geometry formulas reduce the problem to that of
finding any solution in positive integers for the equation
(4641^4) x y z = (880^4) (x+y+z)^3
where the sides of the triangle have lengths x+y, x+z, y+z.
I don't know how to solve such a problem in general. In this case,
since 4641 = (3) (7) (13) (17), we see that (x+y+z) must be divisible
by 4641^2. I optimistically tried x+y+z = 4641^2 = A, so the above
equation becomes
x y z = (4641^2) (880^4) = B
Take x <= y <= z. Then z is a divisor of B, and z lies between A/3 and
A. (You can actually get slightly better bounds on the possible values
of z.) This simple observation eliminates over 95% of the divisors of B
as possible values of z. Try each one, by substituting that value for z
and solving the two equations for x and y. If you get integer
solutions, EUREKA. The values of x, y, z were 141120, 5909761,
15488000
One final question for Jim Buddenhagen: How did you determine that such
a peculiar value (880/4641)^2 would lead to a solvable problem?
-- Barry Wolk <wolk@ccm.UManitoba.CA>