Triangle is Convex Set

Theorem
The interior of a triangle embedded in $\R^2$ is a convex set.

Proof
Denote the triangle as $\triangle$, and the interior of the boundary of $\triangle$ as $\Int \triangle$.

From Boundary of Polygon is Jordan Curve, it follows that the boundary of $\triangle$ is equal to the image of a Jordan curve, so $\Int \triangle$ is well-defined.

Denote the vertices of $\triangle$ as $A_1, A_2, A_3$.

For $i \in \set {1, 2, 3}$, put $j = i \bmod 3 + 1$, $k = \paren {i + 1} \bmod 3 + 1$, and:


 * $U_i = \set {A_i + s t \paren {A_j - A_i} + \paren {1 - s} t \paren {A_k - A_i}: s \in \openint 0 1, t \in \R_{>0} }$

Suppose that the angle $\angle A_i$ between $A_j - A_i$ and $A_k - A_i$ is non-convex.

As $\angle A_i$ is an internal angle in $\triangle$, it follows from definition of polygon that $\angle A_i$ cannot be zero or straight.

Then $\angle A_i$ is larger than a straight angle, which is impossible by Sum of Angles of Triangle Equals Two Right Angles.

It follows that $\angle A_i$ is convex.

From Characterization of Interior of Triangle, it follows that:


 * $\ds \Int \triangle = \bigcap_{i \mathop = 1}^3 U_i$

From Interior of Convex Angle is Convex Set, it follows for $i \in \set {1, 2, 3}$ that $U_i$ is a convex set.

The result now follows from Intersection of Convex Sets is Convex Set (Vector Spaces).