Understanding Thales' Theorem and Its Applications

The first page introduces fundamental concepts of Thales' Theorem and its practical applications in geometry. The content focuses on two main configurations and their applications in calculating lengths and proving parallel lines.

Definition: Thales' Theorem states that when two lines intersect two other lines, the ratios of corresponding segments are equal if and only if the lines are parallel.

Example: In the given configuration, points A, B, D and A, C, E are aligned, with (BC) parallel to (DE). The ratios AB/AD = AC/AE = BC/DE = 0.67 demonstrate parallel lines.

Highlight: Two main applications are presented:

1. Proving lines are parallel by comparing ratios
2. Calculating unknown segment lengths using proportional relationships

Vocabulary:

• Configuration: Specific geometric arrangement of points and lines
• Aligned points: Points that lie on the same straight line
• Parallel lines: Lines that maintain a constant distance and never intersect

Quote: "Les rapports sont égaux, les droites sont donc parallèles" (The ratios are equal, therefore the lines are parallel)

The page includes detailed calculations showing how to find the length of AE = 6 cm using proportional ratios and cross multiplication, demonstrating practical application of the theorem in solving geometric problems.