Proportionality in Geometry: Thales' Theorems, Angle Bisectors, and Power of a Point
Thales' theorem governs parallel transversal proportionality: when two transversals intersect three or more parallel lines, the resulting intercept segments are mutually proportional. We study both the general theorem and its special cases, alongside the angle bisector theorem (internal and external bisectors divide opposite sides proportionally). Expanding proportionality into circle geometry, we analyze intersecting secants, chords, and the power of a point: PM(A) = (AM)² - r². The sign of the power defines whether point A lies outside, on, or inside the circle, with the square root of a positive power yielding the tangent segment length.
1. Thales' & Angle Bisector Theorems
Proportional segments:
- Thales' Theorem: If L1 // L2 // L3, then AB / BC = DE / EF.
- Internal Angle Bisector: In ΔABC with bisector AD: BD / DC = AB / AC.
- Perpendicular Bisectors: The interior and exterior bisectors of the same triangle vertex are mutually perpendicular (angle = 90°).
2. Power of a Point & Secants
PM(A) and circle power:
- Power Formula: PM(A) = (AM)² - r².
• PM(A) > 0 ⇔ A is outside circle, Tangent length = √PM(A).
• PM(A) = 0 ⇔ A is on circle.
• PM(A) < 0 ⇔ A is inside circle. - Intersecting Chords Theorem: EA · EB = EC · ED.
Circle Power Application:
Solution: PM(A) = AM² - r² = 13² - 5² = 169 - 25 = 144.
Tangent length = √144 = 12 cm.
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