Determine if a point lies in a quadrangle [duplicate]

I have a quadrangle which sides consist of parts of rays, and I only know the coordinates of two points on each ray.

I need to determine if a point $$(x,y)$$ lies in such quadrangle.

In this picture, I painted the sides of the quadrangle red in case you don't understand about what quadrangle I'm talking.

marked as duplicate by Dr. Mathva, Lee David Chung Lin, Leucippus, Cesareo, YiFanApr 17 at 13:42

• I multplied LHS of their equations as following. Can you check the sign of this product? And the logic? $$(\frac{x}{5}+ \frac{y}{3}-1).(\frac{x}{4}+ \frac{y}{-7}-1).(\frac{x}{-8}+ \frac{y}{-12}-1).(\frac{x}{5}+ \frac{y}{-2}-1)$$ – Narasimham Apr 17 at 13:51
Let $$A(x_a, y_a), B(x_b, y_b), C(x_c, y_c)$$ and $$D(x_d, y_d)$$ form the quadrilateral $$ABCD$$ with area $$S$$ and let $$P$$ be the point you've chosen.
Calculate the areas of the triangles $$\triangle PAB, \triangle PBC, \triangle PCD$$ and $$\triangle PDA$$, for instance, with the shoelace formula. Let their sum be $$S'$$. Then
\begin{align*} S'=S&\implies P\text{ lies inside }ABCD\\ S'>S&\implies P\text{ lies outside } ABCD \end{align*}