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It's not clear to me why a geometrical proof is so hard to find. The definition says a polygon is convex if we can connect any pair of two points of the polygon with a line that's contained in that polygon. In most sources, for unknown reason, it is treated as an obvious fact.

This is the proof I've found - page 2. Could you verify whether it is correct? I have doubts with regard to this proof - there's a contradiction if we allow angles $> 180$, but how do we know if we can connect two points inside the polygon with a line inside the polygon, if it has angles $<180$? Maybe we would arrive at contradiction if we assumed angles $>170$. We should prove that all angles $< 180$ are okay here.

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  • $\begingroup$ Is there a way to prove the converse of this statement? $\endgroup$
    – user118161
    Apr 2, 2021 at 23:15

2 Answers 2

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Let $A$ be any vertex of the polygon. Let $B$, $C$ be the vertices adjacent to $A$.

Angle $BAC$ (the internal angle) is inside the non-degenerate triangle $BAC$, and hence is less than $180^{\circ}$.

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  • $\begingroup$ Where are you using convexity? $\endgroup$
    – TonyK
    Feb 20, 2015 at 7:34
  • $\begingroup$ Convexity ensures that angle BAC is inside the triangle. $\endgroup$
    – Esteemator
    Feb 20, 2015 at 9:32
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    $\begingroup$ Convexity ensures that angle BAC is inside the triangle. - this is what we are trying to prove here... you cannot assume it. $\endgroup$
    – user216094
    Feb 20, 2015 at 10:33
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    $\begingroup$ I'm not assuming it (as far as I can see). I'm saying that because the polygon is convex, line BC is inside the polygon, and so the angle opposite it (angle BAC) must also be inside the triangle. $\endgroup$
    – Esteemator
    Feb 20, 2015 at 13:52
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Definitions:

$1$. A convex polygon is defined as a polygon with all its interior angles less than $180°$.

$2$. A simple polygon is concave iff at least one of its internal angles is greater than $180°$ .

Let's say an internal angle in a convex polygon is more than $180°$, but it is a contradiction to the definition of a convex polygon.

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