Subject: Re: Trigonometry in a triangle
Date: Fri, 22 Oct 2000 13:14:10 -0300 (GMT)
From: martin lukarevski

Hi Alexander,

Four years ago a young woman posted following problem:

Can this trigonometric inequality be proved

(1) Sin(A)*Sin(B) / Sin2(C/2) + Sin(B)*Sin(C) / Sin2(A/2) + Sin(C)*Sin(A) / Sin2(B/2) ≠ 9,

where A,B,C are angles in a triangle. I wish to say that this one was the hardest but in the same time the most interesting trigonometric inequality I ever come across. Yes, it can be proved but I believe not directly and not by means.

Let's start with the proof. At the beginning we will transform (1) in an equivalent form using formulas:

Sin(A) = a / (2*R) and Sin2(A/2) = (a2 - (b - c )2) / ( 4*b*c )

And analogously for the other terms in (1) where a, b, c are the lenghts of the sides of the triangle and R is its circumradius. With this and using R = a*b*c / ( 4*S ), S being the triangle's area , (1) is transformed to:

(2) 1 / ( a2*( a2 - (b - c)2 ) ) + 1 / ( b2*(b2 - (c - a)2 ) + 1/ ( c2*( c2 - (a - b)2 ) ) 9 / (16*S2)

which is still too difficult to be proved. So, another transformation follows. Now we may put:

x = (b + c - a )/2, y = ( c + a - b )/2, z = ( a + b - c )/2

Hence a = y + z, b = z + x, c = x + y, and x = p - a, y = p - b , z = p - c, where p is the semiperimeter of the triangle.

We must bear in mind that a,b,c are sidelengths and therefore x,y,z are all strictly positive. With this and using Heron's formula

S2 = p*( p - a )*( p - b )*( p - c ),

(2) becomes:

(3) 1 / ( 4*x*y*( x + y )2 ) + 1 / (4*y*z*( y + z )2 ) + 1 / ( 4*z*x*( z + x )2 ) 9 / ( 16*x*y*z*( x + y + z ) )

Eqivalent to (3) is:

(4) K = ( x + y + z )*( x / ( y + z )2 + y / ( z + x )2 + z / ( x + y )2 ) 9 / 4

If we prove (4) for all x,y,z > 0 we are done. For that purpose we will use the following inequality which is standard exercise and can easily be verified:

(*) For all x,y,z > 0, M = x / ( y + z ) + y / ( z + x ) + z / ( x + y ) 3 / 2

And now the last blow. If we use Cauchy-Schwarz-Bunyakovski inequality for the left side of (4) we obtain:

K M2 and M2 9 / 4, so K 9/4.

Q.E.D.

This comletes the proof.

Best regards,
Martin Lukarevski
from Skopje,Macedonia

P.S. I would like to know Alex, what is the source of this problem.

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