Adjacent vertices in convex polygons

[Bipolarity]

While reading a bit about dihedral groups, I encountered a curiosity regarding convex polygons that I'm not sure is true or false.

Given a convex polygon P, let A and B be adjacent vertices of this polygon and let C be a vertex of P not adjacent to A. Then is it necessarily the case that dist(A,B) <= dist(A,C) ?

'<=' is less than or equal to.

For a polygon that is not convex, I found a counterexample to this. For convex polygons, it seems true though I am curious as to how one would prove this, or where one should start.

Thanks!

BiP

 

[jbriggs444]

Consider the polygon defined by vertices A, B, C and D at (1,0), (100,100), (0,1), (0,0).

 

[Bipolarity]

Ah interesting! I thought it was true so didn't proceed to think of counterexamples!
What if we added the restriction that P is a regular polygon? Then I'm pretty sure it's true, but how do we prove it?

Thanks again.

BiP

 

[jbriggs444]

The vertices on a regular polygon will all lie on a circle and will be evenly spaced. The distance between a pair of vertices can be obtained based on the chord of the central angle between the vertices. The central angle and its chord are not difficult to find based on the number of sides between the vertices, the total number of sides of the polygon and the radius of the circle.