# Answer to Question #21809 in Geometry for brennean

Question #21809

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

Expert's answer

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

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

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