All three points lie on a straight line.
The equation for a line is:
(x-x0)/L = (y-y0)/M = (z-z0)/N
where
(L,M,N) = the direction vector for that line.
(x,y,z) = any point on the line.
(x0,y0,z0) = a known point on the line.
So let's create the equation for a line passing through points a and b and see
if point c lies on that line.
(L,M,N) = a - b
(L,M,N) = (2, 5, 3) - (3, 6, 2) = (2 - 3, 5 - 6, 3 - 2) = (-1, -1, 1)
So we have
(x-2)/-1 = (y-5)/-1 = (z - 3)/1
-(x-2) = -(y-5) = (z - 3)
Let's check the points.
a(2,5,3)
-(x-2) = -(y-5) = (z - 3)
-(2-2) = -(5-5) = (3 - 3)
0 = 0 = 0
The above equation is true, so point a lies on the line.
b(3,6,2)
-(x-2) = -(y-5) = (z - 3)
-(3-2) = -(6-5) = (2 - 3)
-1 = -1 = -1
The above equation is true, so point b lies on the line.
c(1,4,4)
-(x-2) = -(y-5) = (z - 3)
-(1-2) = -(4-5) = (4 - 3)
1 = 1 = 1
The above equation is true, so point c lies on the line.
All three points create true expressions for the formula, so all three points lie
on a straight line.
