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### Point within a Triangle

```Date: 05/29/2003 at 13:27:06
From: Venu
Subject: Equilateral Triangles

I have the coordinates of the three corners of a equilateral triangle
ABC. How can I decide whether an arbitrary point (X,Y) lies in the
plane of the triangle?
```

```
Date: 05/29/2003 at 15:29:53
From: Doctor George
Subject: Re: Equilateral Triangles

Hi Venu,

Thanks for writing to Doctor Math.

Let's call your vertices A, B and C. Now assign a direction to the
sides of the triangle by referring to vectors AB, BC, and CA. Note
that the vectors form a loop around the triangle.

Now let's call your point P, define vectors AP, BP, and CP, and
compute the following vector cross products.

AP x AB
BP x BC
CP x CA

If all three of these vectors are in the same direction then P is in
triangle ABC. If P crosses over any side of the triangle then the
cross product using that side will switch direction.

Does that make sense? Write again if you need more help.

- Doctor George, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Linear Algebra
College Triangles and Other Polygons
High School Linear Algebra
High School Triangles and Other Polygons

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