Case 1: Convex, with E the intesect of AC and BD.
Let us fabricate a horizontal table from a frictionless material and drill small holes at the points A,B,C,D into it.
Let us take equal weights and attach strings to them, and put the strings through the holes, and bind together to a point on the upper size of the table, with the weighs hanging down.
We then let the things loose and wait until they finds the equilibrium. The strings will stretch to make straight stages.
On the one hand the weights tend to hang towards the Earth as far as they can, so the sum of the strings hanging down below the table gets maximized, that is, the sum of the lengths on the table gets minimized.
On the other hand the middle point where the strings are stitched together does not move any longer if the forces working on it cancel out.
So for which point E is it true, that the unity vectors (the weigths are equal, so the gravity forces are equal) pointing from E towards A,B,C,D cancel each other out when summed up?
Well, two unity vectors pointing out from a given point yield a sum whose size is proportional to the cosine of the half angle between the directions, and the direction is that of the bisetrix.
From here it follows (I am not not able to provide a figure...) that the angle EA,EB must be equal to the angle EC,ED and the bisectrices must point to opposite directions, which is possible if E is the intesect of AC and BD.
(Huh, that was hard! Not only is English not my first language, arguably not even the second best, and even though I learned and studied maths, never in English.)