The Existence of a circle:
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The Three-Point Rule: There is one and only one circle that can pass through three non-collinear points.
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The Construction: This circle is found by drawing the perpendicular bisectors of the line segments joining the points. The point where these bisectors intersect is the center (\(O)\).
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Circumcircle: When this circle passes through the vertices of a triangle, it is the circumcircle, its center is the circumcentre, and the distance to any vertex is the circumradius \((OX = OY = OZ)\).
Chord-center Relationship:
A chord’s geometry is defined by its relationship to the center \(O\) through perpendicularity and distance.
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Bisection & Perpendicularity: A perpendicular line from the center to a chord bisects the chord \((PR = RQ)\).
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Converse: The line joining the center to the midpoint of a chord is perpendicular to it \((90^\circ)\).
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Equidistance: Equal chords are always at an equal distance from the center. Conversely, chords that are the same distance from the center are equal in length.
Angular properties of chords:
Chords "project" angles toward the center based on their length.
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Equal Chords, Equal Angles: If two chords are equal in length, they subtend equal angles at the center \((\angle POQ = \angle ROS)\).
Converse: If the angles subtended by two chords at the center are equal, then the chords themselves are equal.
The calculation Framework (Pythagoras):
Most numerical problems involving chords use the Pythagorean Theorem because the radius, the distance from the center, and the half-chord form a right-angled triangle.
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Theorem Application: \(OP^2 = OR^2 + RP^2\)
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Key variables:
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\(OP\) = Radius (\(r\))
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\(OR\) = Perpendicular distance from center (\(d\))
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\(RP\) = Half of the chord length (\(\frac{c}{2}\))
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The Core Theorem: Opposite Angles:
The most important property is that the opposite angles of a cyclic quadrilateral are supplementary, meaning they always add up to \(180^\circ\)
Important!
Rule: \(\angle A + \angle C = 180^\circ\) and \(\angle B + \angle D = 180^\circ\)
The Exterior Angle Property:
If you extend one side of a cyclic quadrilateral outward to create an exterior angle, a unique relationship appears:
Theorem: The exterior angle is exactly equal to the interior opposite angle
Logic: Since the exterior angle and its neighbor on the line add to \(180^\circ\), and the opposite interior angle also adds to \(180^\circ\) with that same neighbor, they must be equal to each other.
Cyclic Quadrilateral
To prove a quadrilateral is cyclic (meaning a circle can be drawn through all four corners), you just need to show one of the following:
- One pair of opposite angles adds to \(180^\circ\)
- An exterior angle is equal to its interior opposite angle.
- The vertices are concyclic (they all lie on the same path around a center \(O\)).
Comprehensive Circle Theory Table:
| Category | Theorem | Key Property |
| Points & Circles | The Unique Circle | Only one circle can pass through 3 non-collinear points. |
| Triangles | Circumcircle | The circle passing through all vertices of a triangle. Center \(O\) is the intersection of perpendicular bisectors. |
| Chord Bisection | Perpendicular bisector | A line from the center perpendicular to a chord bisects it (divides it in half). |
| Chord Angles | Equal Angles | Equal chords subtend (create) equal angles at the center. |
| Chord Distance | Equidistance | Equal chords are the same distance from the center. |