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

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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

Point Inside Triangle.png

From External Angle of Triangle 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 theorem is Proposition $21$ of Book $\text{I}$ of Euclid's The Elements.


Sources