Set up some coordinate system such that, in this coordinate system, the points are given by $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$. Let the given point be "(x*, y*)". Now determine whether the given point is on the same side of each line as the third vertex:
1) Find a, b such that both $(x_1, y_1)$ and $(x_2, y_2)$ lie on line y= ax+ b (in other words, find the equation of the line determined by those two points). Determine if $y_3- ax_3- b$ and $y^*- ax^*- b$ have the same sign. If not we are done and the answer is "no, the point (x*, y*) does not lie inside this triangle. If "yes" then
2) Find a, b such that both $(x_1, y_1)$ and $(x_3, y_3)$ lie on line y= ax+ b. Determine if $y_2- ax_2- b$ and $y^*- ax^*- b$ have the same sign. If not we are done and the answer is "no", the point (x*, y*) does not lie inside this triangle. If "yes" then
3) Find a, b such that both $(x_2, y_2)$ and $(x_3, y_3)$ lie on line y= ax+ b. Determine if $y_1- ax_1- b$ and $y^*- a^x*- b$ have the same sign. If not we are done and the answer is "no", the point (x*, y*) does not lie inside this triangle. If "yes" then point (x*, y*) does lie inside the triangle.
Note: if, for i= 1, 2, or 3, $y_i- ax_i- b$ is 0 then the three points do not define a triangle. If, for any i= 1, 2, or 3, $y^*- ax^*- b$ is 0, the points lies on the triangle, not inside it.