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.

Corollary

The angle between the two line segments to the point inside the triangle is greater than the angle between the other two sides of the triangle.

Proof

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

$\blacksquare$

Proof of Corollary

$\angle BDC > \angle CED$ because an external angle of a triangle is greater than either opposite internal angle.

Similarly, $\angle CEB > \angle BEC$.

Since $\angle CED$ is the same angle as $\angle CEB$, $\angle BDC > \angle CEB > \angle BEC$.

$\blacksquare$

Historical Note

This is Proposition 21 of Book I of Euclid's The Elements.