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

Theorem

The angle between the two line segments from the endpoints of one side to a point inside the triangle is greater than the angle between the other two sides of the triangle.

In the words of Euclid:

If on one of the sides of a triangle, from its extremities, there be constructed two straight lines meeting within the triangle, the straight lines so constructed will be less than the remaining two sides of the triangle, but will contain a greater angle.

(The Elements: Book $\text{I}$: Proposition $21$)

Proof

From External Angle of Triangle is Greater than Internal Opposite:

$\angle BDC > \angle CED$

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 proof is Proposition $21$ of Book $\text{I}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 1 (2nd ed.) ... (previous) ... (next): Book $\text{I}$. Propositions