In a triangle ABC two items are defined : the side AB=c and the height from point C to this side H_c= k.

What is the maximum value of the product H_a*H_b* H_c ?

S=Area =c*k/2
As  Hc=c is given, we need to maximize Ha*Hb :
Ha=2*S/a
Hb=2*S/b
Denoting the angle between a and b as Gamma, we have the expression for the triangle area as :
S=a*b*Sin(Gamma)/2
which leads to :
a*b=2*S/Sin(Gamma)
from the above:
Ha*Hb=4*S^2/(a*b)=2*S*Sin(Gamma)
The max. of Ha*Hb is therefore : 2*S=c*k
and the max for Ha*Hb*Hc will be :

                                                    k*c^2

Posted by Dan Rosen on 2013-09-22 13:36:57