Lecture on Euclidean Geometry: The Nine Theorems

Introduction

• The lecture covers nine theorems of Euclidean geometry related to circles.
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Theorem 1: Line from Center Perpendicular to Chord

• Statement: If a line is drawn from the center of a circle and is perpendicular to a chord, it bisects the chord.
  • Given: Line OC ⊥ AB (chord), then AC = CB.
• Converse: If a line from the center bisects a chord, it is perpendicular to the chord.
  • Given: AC = CB, then OC ⊥ AB.
• Reason Stated: Line from center perpendicular to chord.

Theorem 2: Angle at the Center

• Statement: The angle at the center of a circle is twice the angle at the circumference.
  • Angle AOC = 2 * Angle ABC.
• Example: If angle ABC = 40°, then angle AOC = 80°.
• Conditions: O must be the center, and B must be on the circumference.
• Reason Stated: Angle at center equals 2 * angle at circumference.

Theorem 3: Angle in a Semicircle

• Statement: An angle subtended by a diameter is 90°.
  • Example: If AC is a diameter, then angle ABC = 90°.
• Reason Stated: Angle at semicircle.

Theorem 4: Opposite Angles of a Cyclic Quadrilateral

• Statement: Opposite angles of a cyclic quadrilateral are supplementary (sum to 180°).
  • Angle A + Angle C = 180° and Angle B + Angle D = 180°.
• Reason Stated: Opposite angles of a cyclic quad.

Theorem 5: Exterior Angle of Cyclic Quadrilateral

• Statement: The exterior angle of a cyclic quadrilateral is equal to the opposite interior angle.
• Example: Angle BDE = Angle CAB.
• Reason Stated: Exterior angle of a cyclic quad.

Theorem 6: Angles Subtended by the Same Chord

• Statement: Angles subtended by the same chord or arc are equal.
  • Example: Angle DAC = Angle DBC.
• Reason Stated: Angle at same segment.

Theorem 7: Angles Subtended by Equal Chords

• Statement: Angles subtended by equal chords are equal.
  • Given: BC = DE, then Angle BAC = Angle DFE.
• Reason Stated: Angles subtended by equal chords.

Tangents

Theorem 8: Tangent Perpendicular to Radius

• Statement: A tangent to a circle is perpendicular to the radius at the point of tangency.
  • Example: OB ⊥ AC.
• Reason Stated: Tangent perpendicular to the radius.

Theorem 9: Tangents from a Common External Point

• Statement: Tangents drawn from the same external point to a circle are equal.
  • Example: AP = BP.
• Reason Stated: Tangents from the same point.

Theorem 10: Tangent-Chord Theorem

• Statement: The angle between a tangent and a chord is equal to the angle in the opposite segment.
  • Example: Angle DBA = Angle DCA.
• Reason Stated: Tan chord theorem.

Conclusion

• The lecture covers the essential theorems necessary for understanding circle geometry.
• Future sessions will apply these theorems to solve problems.
• Ensure to be familiar with theorems for proofs and solving geometry questions.

This summary provides a structured overview of the key points and theorems discussed in the lecture on Euclidean Geometry.