The interior of the triangle is the set of all points inside a triangle, i.e., the set of all points in the convex hull of the triangle's vertices.
The simplest way to determine if a point lies inside a triangle is to check the number of points in the convex hull of the vertices of the triangle adjoined with the point in question. If the hull has three points, the point lies in the triangle's interior; if it is four, it lies outside the triangle.
To determine if a given point 
lies in the interior of a given triangle, consider an individual
vertex, denoted 
,
and let 
and 
be the vectors from 
to the other two vertices. Expressing the vector from 
to 
in terms of 
and 
then gives
 |
(1)
|
where 
and 
are constants. Solving for 
and 
gives
 |  |  |
(2)
|
 |  |  |
(3)
|
where
 |
(4)
|
is the determinant of the matrix formed from the column vectors 
and 
. Then the point 
lies in the interior of the triangle if 
and 
.
If the convex hull of the triangle vertices plus the point 
is bounded by four points, the point 
lies outside the triangle. However, if it contains three
points, the point 
may lie either in the interior or in the exterior.
