Geometric Proofs

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Things you are assumed to know already

• Angle sum theorems for triangles and quadrilaterals
• Properties of intersecting lines
  • Congruence of vertically opposite angles
• Relationships involving angles in parallel lines with a transversal
  • Corresponding, alternate, interior
• Isosceles angle theorem
  • Angles opposite to the equal sides of an isosceles triangle are also equal.
• Triangle congruence and similarity “tests”
• In a triangle, an exterior angle is the sum of the 2 remote interior angles

Circles

• Circle: set of all points at a given distance $r$ from a given point $O$ called the centre

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Important Circle Theorems

1. The angle at the centre is twice the angle at the circumference
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2. Angles in a semicircle are right angles
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3. Angles in the same segment are equal
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4. Opposite angles in a cyclic quadrilateral are supplementary
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5. Angle between a tangent and the radius drawn at the point of contact is a right angle

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6. Angle between a tangent and chord is equal to the angle in the alternate segment
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7. The 2 tangents drawn from a point to a circle are of equal length

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8. The products of the intercepts of 2 intersecting chords are equal
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9. The square of the length of the tangent from an external point is equal to the product of the intercepts of the secant passing through this point
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