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MATHEMATICS THEOREM
(TWO THEOREM EVERY 2 WEEKS)
THEOREM 1:Ptolemy theorem

Fig 1.
In Fig 1, ABxCD+BCxAD=ACxBD
Proof:

Fig 2.
Triangle ABE ~Triangle ACD
so AB/BE=AC/CD,
    ABxCD=BExAC------------(!)
Triangle AED~Traiangle CBA
so DE/BC=AD/AC,
    DExAC=ADxBC------------(#)
(!)+(#):
    BExAC+DExAC=ABxCD+ADxBC
So,BDxAC=ABxCD+ADxBC

THEOREM 2: Cauchy's Inequality
Let a1,a2,a3,.....,an and b1,b2,b3,.....,bn be real numbers,then
(a1b1 + ... + anbn)^2 (a1^2+ ... + an^2) *( b1^2 + ... + bn^2)

Proof:
Sum of (aix+bi )^2
= (a1^2+ ... + an^2)x^2+(a1b1 + ... + anbn)x+(b1^2+... +bn^2)--------(!)
As discriminant of (!) must not be negative,
so discriminant of (!) must be smaller or equal to zero.
It's the Cauchy's inequality.