State and prove Basic Proportionality theorem.

Step 1: State the Basic Proportionality Theorem

The Basic Proportionality Theorem, also known as Thales Theorem, describes a fundamental property of triangles when a line parallel to one of its sides intersects the other two sides. It states that the line divides the other two sides proportionally.

Step 2: Setup for Proof

Consider a triangle $ABC$. Let $DE$ be a line segment drawn parallel to side $BC$, intersecting sides $AB$ at $D$ and $AC$ at $E$. We need to prove that the ratio of the segments $AD$ to $DB$ is equal to the ratio of $AE$ to $EC$.

Step 3: Construction

To prove the theorem, we make two constructions. First, we join $B$ to $E$ and $C$ to $D$. Second, we draw perpendiculars from $E$ to $AB$ (let's call the foot of the perpendicular $M$) and from $D$ to $AC$ (let's call the foot of the perpendicular $N$). These perpendiculars will be used to calculate the areas of the triangles.

Step 4: Calculate Area Ratios

The area of a triangle is given by half times base times height. Using $EM$ as the height for triangles $ADE$ and $BDE$, we find the ratio of their areas is $AD/DB$. Similarly, using $DN$ as the height for triangles $ADE$ and $CDE$, the ratio of their areas is $AE/EC$.

Step 5: Equate Area Ratios

Triangles $BDE$ and $CDE$ lie between the same parallel lines $DE$ and $BC$, and they share the same base $DE$. Therefore, their areas are equal. By substituting this equality into our previous area ratios, we conclude that $AD/DB$ must be equal to $AE/EC$, thus proving the Basic Proportionality Theorem.

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