Answer to Question #21809 in Geometry for brennean

Question

if given 4 non-collinear points,how do you decide if its a rectangle?

Answer 1

For example 4 points are:
Ai (xi, yi, zi) i=1,2,3,4
1) this 4 points are in the same plane.
it mean what triple product (A1A2, A1A3, A1A4)= (A1A2*(A1A3 x A1A4)) = 0
AiAj = (xi - xj, yi-yj, zi - zj)

2) if A1A2A3A4 is a rectangle it is possible tofind i, j, k, l
AiAj*AiAk=0 - (* - is a dotproduct)
AiAj*AiAk = (xi - xj)*(xi - xk) +(yi-yj)*(yi-yk)+ (zi - zj)*(zi - zk)
It is mean what the angle between AiAj and AiAk equal 90.
for others the same:
AkAl*AkAi=0
AkAl*AlAj=0