let A1, A2, A3 ......An, be the vertices of an n sided regular polygon inscribed in a circle of radius R. If (A1A2)^2+ (A1A3)^2 +.......+ (A1An)^2 = 14(R)^2. Find n.

e know that for a polygon(a = side lenght and n = no of sides), Radius of circumcircle of this polygon =

R = a/2sin(pi/n)

so a = 2Rsin(pi/n) 

so A1A2=2Rsinα, A1A3=2Rsin(2α) and A1A4=2Rsin(3α), 
where α=π/n. 
now as per given condition
1/(2Rsinα) = 1/[2Rsin(2α)] + 1/[2Rsin(3α)] 
1/sinα = 1/sin(2α) + 1/sin(3α) 
1/sin(2α) = 1/sinα - 1/sin(3α) 
1/sin(2α) = [sin(3α) - sinα] / [sinα * sin(3α)] 
1/sin(2α) = 2sinα * cos(2α) / [sinα * sin(3α)] 
1/sin(2α) = 2cos(2α) /sin(3α) 
sin(3α) = 2sin(2α)*cos(2α) 
sin(3α) = sin(4α)  
3α + 4α = π 
7α = π 
7π/n = π 
n=7