Proof: Tangent-Secant Theorem (Product of Secant Segments Equals Tangent Squared)

The Tangent-Secant Theorem

• 🎯 The video aims to prove that for a circle, the product of the lengths of a secant segment and its external segment is equal to the square of the length of the tangent segment that intersects it.
• 🔑 This is mathematically stated as: AC * AB = AD², where AC is the secant segment, AB is its external part, and AD is the tangent segment from the intersection point A.

Geometric Proof Strategy

• 🔬 The proof relies on constructing similar triangles within the circle's geometry.
• 💡 Two key triangles, △ABD and △ADC, are identified and shown to be similar.

Establishing Triangle Similarity

• 📐 The similarity is established using the Angle-Angle (AA) similarity postulate.
• 1️⃣ The angle at vertex A (∠BAC) is common to both triangles.
• 2️⃣ The inscribed angle subtending a specific arc is equal to the angle formed by a tangent and a chord that subtend the same arc. Specifically, ∠BDA (angle between tangent AD and chord BD) is equal to ∠ACD (inscribed angle subtending arc BD).
• 3️⃣ Since two angles are congruent, the third angles must also be congruent, confirming △ABD ~ △ADC.

Deriving the Theorem

• 📈 From the similarity of △ABD and △ADC, we can set up a proportion between corresponding sides.
• ⚖️ The ratio of the longest side to the shortest side in △ADC (AC/AD) is equal to the ratio of the longest side to the shortest side in △ABD (AD/AB).
• 🧮 Cross-multiplying this proportion (AC * AB = AD * AD) leads directly to the theorem: AC * AB = AD².