# Determine if a point is in a triangle? [duplicate]

Given the points of the vertices of a triangles as tuples (x,y) and a point P=(x,y).

How can I determine if this point P is contained in the triangle (assume that it's not on the border of the triangle)?

Given three non-collinear points in $\mathbb{R}^2$ (the vertices of a triangle) $A,B,C$ and a point $P$, there is a unique way to represent $P$ as $$P=\lambda_A A + \lambda_B B + \lambda_C C$$ with $\lambda_A,\lambda_B,\lambda_C$ being real coefficients fulfilling $\lambda_A+\lambda_B+\lambda_C=1$. This kind of representation is also known as exact barycentric coordinates. The coefficients $\lambda_A,\lambda_B,\lambda_C$ are straightforward to find through linear algebra and the point $P$ strictly lies on the interior of $ABC$ iff $$\lambda_A>0,\quad\lambda_B>0,\quad\lambda_C>0,$$ that is equivalent to $$\begin{pmatrix}1 & 1 & 1 \\ x_A & x_B & x_C \\ y_A & y_B & y_C\end{pmatrix}^{-1} \begin{pmatrix}1 \\ x_P \\ y_P\end{pmatrix}>0.$$
An equivalent alternative, assuming that $A,B,C$ are counter-clockwise ordered, is to compute the (oriented) areas of $ABP,BCP,CAP$ through the shoelace formula and check that all these (oriented) areas are positive. 