Lines Through Endpoints of One Side of Triangle to Point Inside Triangle is Less than Sum of Other Sides

Theorem
Given a triangle and a point inside it, the sum of the lengths of the line segments from the endpoints of one side of the triangle to the point is less than the sum of the other two sides of the triangle.

Proof

 * Point Inside Triangle.png

Given a triangle $ABC$ and a point $D$ inside it.

We can construct lines connecting $A$ and $B$ to $D$, and then extend the line $AD$ to a point $E$ on $BC$.

In $\triangle ABE$, $AB + AE>BE$.

Then, $AB + AC = AB + AE + EC > BE + EC$ by Euclid's second common notion.

Similarly, $CE + ED > CD$, so $CE + EB = CE + ED + DB > CD + DB$.

Thus, $AB + AC > BE + EC > CD + DB$.