Can Point inside quad be determined with angles alone?

Question

Given

A Quad($C$, $D$, $E$, $B$) and Points $A$, $G$, $F$

Question

Is it possible by calculating the angles between points to determine whether a point is inside (including on), or outside the quad

Observations:

Off the top of my head it looks like

$\angle CFB + \angle CFD + \angle DFE + \angle EFB = 360 ^{\circ} $

$\angle CGD = 180 ^{\circ}$ and the other angles: $\angle CGB + \angle BGE + \angle EGD = 180 ^{\circ} $

Is this a valid way to test whether a point lies inside a quad? I can't seem to come up (in my head) with a situation in which the angles from point A would total $360 ^{\circ}$ as well

Side Note:

I'm hoping to use this information to turn this into a little software routine in python (which it now occurs to me might be subject to rounding errors...)

Answer 1

Don't know if you can do it just with angles (e.g. what if the quadrilateral has a reflex angle in it?) But any quadrilateral can be split into two triangles (e.g. BCD and BDE for your notation). Then your problem reduces to determining if a point is an interior or boundary point of either of the triangles.

How to determine some points are inside or outside in triangle