Show 3 pts are collinear formula - Can Someone Explain it to me? (1 Viewer)

[~shinigami~]

To show that 3 points are Collinear you can use:

x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂) = 0

It works and I can remember this easily and it seems so much easier but I would like someone to explain to me the theory behind it incase I ever have argue that this method is valid to a teacher.

The website I got it off explains it as "A slightly more tractable condition is obtained by noting that the area of a triangle determined by three points will be zero iff they are collinear" but I'm still confused about the theroy.

Also how would I set my "working out" so that the marker would get what I was trying to do already.

Thank You in advance to all you smart people.

 

[PC]

Why don't you just take two points and get the equation of that straight line. Then substitute the third point into it and show that it works? Always make life easy for the markers. After all, they're the ones giving you your marks!

I've never seen it before, but I suppose it goes something like this:

Two point form of the equation of a straight line is:
(y₂ – y₁)/(x₂ – x₁) = (y – y₁)/(x – x₁)
Multiplying to eliminate denominators:
(x – x₁)(y₂ – y₁) = (x₂ – x₁)(y – y₁)

Now since the third point should be on the same straight line, its co-ordinates must satisfy the equation. Substituting gives:
(x₃ – x₁)(y₂ – y₁) = (x₂ – x₁)(y₃ – y₁)
x₃(y₂ – y₁) – x₁(y₂ – y₁) = x₂(y₃ – y₁) – x₁(y₃ – y₁)
x₃y₂ – x₃y₁ – x₁y₂ + x₁y₁ = x₂y₃ – x₂y₁ – x₁y₃ + x₁y₁
x₁y₂ – x₁y₃ – x₂y₁ + x₂y₃ + x₃y₁ – x₃y₂ = 0
x₁(y₂ – y₃) – x₂(y₁ – y₃) + x₃(y₁ – y₂) = 0
x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂) = 0

Good luck remembering where everything goes!

 

[~shinigami~]

Thanks for explaining it guys.

But like you said, I'll just stick to the normal method.