A common technique in a computer program is to use barycentric coordinates.
Barycentric coordinates are a lot easier to find than any web resources indicate, so I'm not linking to them.
The easiest way to obtain barycentric coordinates of a point P, given a triangle with vertices described by the vectors A, B, C is likely this method:
$ AreaABC = \frac{ \left| \overline{AB} \times \overline{AC} \right| }{ 2 } $
$ \alpha = \frac{ \left| \overline{PB} \times \overline{PC} \right| }{ 2AreaABC } $
$ \beta = \frac{ \left| \overline{PC} \times \overline{PA} \right| }{ 2AreaABC } $
$ \gamma = 1 - \alpha - \beta $
Here $\alpha$ is the ratio of the area of a subtriangle PBC over the area of the whole triangle ABC, as shown in this image from Peter Shirley's book:

If ALL of $ 0 \le \alpha \le 1, 0 \le \beta \le 1, 0 \le \gamma \le 1 $, then the point P is inside the triangle.
If ANY of $\alpha$,$\beta$,$\gamma$ are outside those ranges, then the point P is not inside the triangle.
Note also when one of $\alpha$,$\beta$,$\gamma$ is 0, and the other 2 coordinates are between 0 and 1, the point P is on an edge of the triangle.
When one of $\alpha$,$\beta$,$\gamma$ is 1 and the other two are 0, then the point P is exactly at a vertex of the triangle.
Of course, these computations assume P is already in the plane of the triangle. If P is not in the plane of the triangle, then you should project it there first, before computing the barycentric coordinates.