Let $ABC$ be a triangle. We construct the square $BCDE$ outside the triangle $ABC$ and the square $ACGF$ outside the triangle $ABC$. We denote $M$ and $N$ as the midpoints of the segments $[FD]$ and $[GE]$. Calculate the length $MN$ in terms of the lengths of the sides of the triangle $ABC$.
We introduce $I$ as the point such that the quadrilateral $I D C G$ is a parallelogram. Then the lines $(G I),(C D)$, and $(B E)$ are parallel. Therefore, by alternate interior angles, $\widehat{N E B}=\widehat{N G I}$ and $N E=N G$ and $B E=G I$. Thus, the triangles $N E B$ and $N G I$ are congruent. Therefore, the points $I, N$, and $B$ are collinear. Similarly, we obtain that the points $I, M$, and $A$ are collinear. Furthermore, the points $M$ and $N$ are the midpoints of the segments $[A I]$ and $[B I]$. Therefore, by the theorem of Thales, $M N=\frac{1}{2} A B$.

## Ceva's Theorem

The second, more significant step towards projective geometry is the following theorem, known as Ceva's Theorem:

Theorem 1. Let $A B C$ be a triangle and $X, Y$, and $Z$ be points on the sides $[B C],[C A]$, and $[A B]$, respectively. Then the lines $(A X),(B Y)$, and $(C Z)$ are concurrent if and only if

$$
\frac{X B}{X C} \cdot \frac{Y C}{Y A} \cdot \frac{A Z}{Z B}=1
$$

![](https://cdn.mathpix.com/cropped/2024_05_10_6d86e9a87432417d5c97g-211.jpg?height=623&width=719&top_left_y=1125&top_left_x=817)

Proof.

Let $W$ be the intersection of the lines $(C Z)$ and $(B Y)$, and let $X^{\prime}$ be the intersection of the lines $(A W)$ and $(B C)$. It suffices to show that

$$
\frac{X^{\prime} C}{X^{\prime} B}=\frac{Y C}{Y A} \cdot \frac{A Z}{Z B}
$$

indeed, the lines $(A X),(B Y)$, and $(C Z)$ are concurrent if and only if $X=X^{\prime}$.

We draw the line $d$ parallel to the line $(B C)$ passing through the point $A$. We denote by $U$ and $V$ the points of intersection of the lines $(B Y)$ and $(C Z)$ with the line $d$, respectively.

By the theorem of Thales in the "butterfly" $A V W C X^{\prime}, \frac{X^{\prime} C}{A V}=\frac{X^{\prime} W}{A W}$. Similarly, in the butterfly $A U V W B X^{\prime}, \frac{X^{\prime} B}{A U}=\frac{X^{\prime} W}{A W}$. Combining these, we get

$$
\frac{X^{\prime} C}{X^{\prime} B}=\frac{A V}{A U}
$$

On the other hand, the theorem of Thales in the butterfly $A V Z C B$ gives $\frac{A V}{C B}=\frac{A Z}{Z B}$. Similarly, in the butterfly $A U Y B C$, we find $\frac{A U}{C B}=\frac{A Y}{Y C}$. Combining these two ratios, we get:

$$
\frac{X^{\prime} C}{X^{\prime} B}=\frac{A V}{A U}=\frac{A V}{B C} \cdot \frac{B C}{A U}=\frac{A Z}{Z B} \cdot \frac{C Y}{A Y}
$$

which gives the desired result.

Ceva's Theorem allows us to prove the existence of many special points.