Point inside a Triangle

Date: 07/15/2001 at 14:53:58
From: Dingo Bob
Subject: Checking to see if a point is inside a triangle

If you have any four points, and three of them are the corners of a 
triangle, is there a formula to determine whether or not the fourth is 
inside the triangle? If not, what's the best way to check? The three 
corners can be anywhere (positive or negative) and the triangle is not 
necessarily (but can be) anything special (isoceles, equilateral, 
etc.).

Thank you for reading my question.

Date: 07/17/2001 at 17:26:20
From: Doctor Jaffee
Subject: Re: Checking to see if a point is inside a triangle

Hi Dingo Bob,

The easiest way to determine whether a point is inside a triangle is 
to plot the points on a grid, connect the vertices of the triangle, 
and look and see. However, if I understand your question correctly, 
you are looking for a strictly algebraic method for solving the 
problem.

Here is how I would do it.

First, I would pick any two of the vertices and determine the equation 
of the line that contains them. For example, suppose the three 
vertices are at A(-2,3), B(5,1) and C(-6,-4). The equation of the line 
that connects A and B is y = (-2/7)x + 17/7. 

If you substitute the cooordinates of point C into the equation, you 
get -4 = (-2/7)(-6) + 17/7, or -4 = 29/7, but -4 is much smaller than 
29/7, so we can conclude that C is below the line that connects A and 
B. That means that the fourth point (let's call it D) must be below 
the line that connects A and B.

Next, find the equation of the line containing B and C. You will see 
that A is above that line, so D must be above the line.  

Finally, find the equation of the line containing A and C. You will 
find that B is below that line, so D must be below the line.

If D satisfies all three conditions (below the two lines and above the 
other), then D is inside the triangle; otherwise it isn't.

I hope my answer has helped you understand the problem better. I have 
assumed that you understand how to find the equation of a line, given 
two points on the line. If you don't, check the archives for help or 
write back. You might also want to read a more general answer from our 
archives:

   Formula for Point in Rectangle
   http://mathforum.org/dr.math/problems/scott5.31.96.html   

Thanks for writing to Ask Dr. Math.

- Doctor Jaffee, The Math Forum
  http://mathforum.org/dr.math/