Consider eight points that are equally spaced on a circle in which the length of the radius is 1. Find the product of the distances from one of the points to each of the other seven points.
We can let the eight points be
A = (1,0) B = (1/√2 , 1/√2) C = (0, 1) D = (-1/√2, 1/√2) E = (0, -1)
F = (-1/√2, -1/√2) G = (0, -1) H = (1/√2, -1/√2)
Let A be the selected point that we will use to calculate the distances to every other point
We can use the distance formula to calculate the distance from A to the other seven points [which you should be able to work with ]
I will use WolframAlpha to calculate these
AB = √ [ 2 -√2]
AC = √2
AD = √[2 + √2 ]
AE = 2
AF = √[2 + √2 ]
AG = √2
AH = √[2 - √2]
And computing these products we have
2 * √2 * √2 * (√[2 - √2])^2 * ( √[2 + √2] )^2 =
2 * 2 * ( 2 - √2) ( 2 + √2) =
4 * [ 4 - 2 ] =
4 * 2 =
8