Euclidean Geometry
Circle theorems, proportional intercept theorem
Circle Theorems and Proportional Intercept Theorem
Grade 12 extends circle geometry with more complex proofs including:
• Tangent-chord theorem: angle between tangent and chord = angle in alternate segment
• Proportional intercept theorem: a line parallel to one side of a triangle divides the other two sides proportionally
Example
Tangent-Chord Theorem
A tangent at point P meets chord PQ.
Angle between tangent and PQ = angle in alternate segment.
If the tangent-chord angle = 55°, then the angle subtended by PQ in the alternate segment = 55°.
Example
Proportional Intercept (Midpoint Theorem Extended)
In △ABC, D is on AB and E is on AC such that DE ∥ BC.
Then AD/DB = AE/EC.
If AD = 4, DB = 6, AE = 3:
AE/EC = AD/DB → 3/EC = 4/6
EC = 3 × 6/4 = 4.5
Note
Remember
In Euclidean geometry proofs, always state the theorem or reason for each step. Draw clear diagrams and mark given information. Proportional intercept theorem requires the line to be parallel to a side of the triangle.
Key Vocabulary
Tangent-chord angleThe angle between a tangent and a chord drawn from the tangent point
Alternate segmentThe segment on the opposite side of the chord from the tangent
ProportionalHaving the same ratio
ParallelLines that are always the same distance apart and never meet
ProofA logical argument showing a statement is true
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Tangent-chord angle
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