Proving that a line is straight

[daniel_i_l]

In various riddles and math questions you're asked to prove that 3 points are on the same line. what theorms can be used to prove this sort of thing. in other words, what properties of the points can be used to prove that they're all on a line?
Thanks.

 

[d_leet]

Choose two of them find the slope of those two points, pick two others(one of the originals and the other one) and find the slope between those two, if the slopes are the same they lie on the same line.

 

[Integral]

Choose two of them find the slope of those two points, pick two others(one of the originals and the other one) and find the slope between those two, if the slopes are the same they lie on the same line.

Ah... No.

Parallel lines have the same slope, but never intersect.

Find the equation of a line between 2 of the points.

Do the other points satisfy the equation? If so, they are on the line.

 

[AlephZero]

Some other properties that might be useful:

Area of the triangle defined by the 3 points = 0

Vector cross product of two vectors joining the points = 0

If the points have position vectors a b and c in order along a line, then b = ka + (1-k)c for some constant k with 0 < k < 1.

 

[AlephZero]

Parallel lines have the same slope, but never intersect.

True, but in d_leet's method the two lines always intersect because they have a common point.