Suppose ABC is an equilateral triangle and P is a point inside the triangle, such that PA = 3 cms., PB = 4 cms., and PC = 5 cms.
Then find the length of the side of the equilateral triangle.
s=sqrt(25+12sqrt(3))
Proof:
s=side of triangle
a=angle PCB
in PCB and PAC  theorem of cosinus:
16=25+s²10s(cos(a))
9=25+s²10s(cos(pi/3a))
s²10s(cos(a))+9=0
s²5s(cos(a))5s√3(sin(a))=0
eliminate a
cos(a)=(s²+9)/(10s)
sin(a)=sqrt(1cos(a)*cos(a))
the result is an equation (bisquare):
s^4  50s^2 + 193 = 0
the unique solution satisfying the side of a triangle is:
s=sqrt(25+12√3)

Posted by luminita
on 20040108 08:44:39 