Let ABC be an equilateral triangle and P, a point in

the plane of ABC, not lying on any of the lines AB,

BC, or CA.

Let points A', B', and C' lie on lines BC, CA, and AB

respectively such that |AA'| = |BB'| = |CC'|.

Let points A", B", and C" lie on lines BC, CA, and

AB respectively such that PA" || AA', PB" || BB',

PC" || CC'.

Prove that

|AA'| = S

_{A}*|PA"| + S

_{B}*|PB"| + S

_{C}*|PC"| ,

where S

_{X}= +1 if P and vertex X lie on the same side

of the sideline opposite vertex X,

= -1 otherwise.