2
$\begingroup$

I'm really not sure where to start. Induction can really be used, and that seems like the only way to prove for all $n$.

$\endgroup$
2
$\begingroup$

I will construct an explicit series of polygons satisfying the conditions. No induction needed.

  • Three consecutive vertices of the polygon $A,B,C$ define an equilateral triangle.
  • The remaining $n-3$ vertices lie along the arc between $A$ and $C$ with centre $B$ and are joined in sequence.

$\angle B$ is 60° by definition, while $\angle A$ and $\angle C$ are acute because they are the angles between a chord and a radius. For each other vertex, its neighbours form a chord that lies inside the $AC$ chord, so its angle must be greater than 120° and thus obtuse. All angles are less than 180°, so the polygon is convex and the proof is finished.

$\endgroup$
1
$\begingroup$

You might start with an equilateral triangle, and put the other $n-3$ vertices on an arc of a circle joining two vertices of the triangle and centred on the third.

enter image description here

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.