It’s time for a brain teaser:

There is a triangle , and is an arbitrary interior point of this triangle (see Fig. 1). Prove that .

Fig. 1

Here is my solution:

Extend line to point on (see Fig. 2),

Fig. 2

we have

My solution relies on a well known theorem:

Given a triangle ABC, the sum of the lengths of any two sides is greater than the length of the third side.

In the words of Euclid:

“In any triangle two sides taken together in any manner are greater than the remaining one” (The Elements: Book I: Proposition 20)

I have conjured up the following algebraic proof of Euclid’s proposition:

Any can be put in a rectangular coordinate system where (see Fig. 3)

Fig. 3

It follows that