Circle Theorems

Why This Matters

Circle theorems form the backbone of geometric reasoning about one of mathematics' most fundamental shapes. You're being tested on your ability to recognize angle-arc relationships, power of a point, and perpendicularity conditions—not just memorize formulas. These theorems connect algebraic manipulation with geometric intuition, appearing repeatedly in proofs, coordinate geometry problems, and optimization questions.

The beauty of circle theorems lies in how they interlock: the Inscribed Angle Theorem leads to Thales' Theorem, which connects to cyclic quadrilaterals, which relies on the same arc-angle principles. Master the underlying mechanisms—why an inscribed angle is half its arc, how the power of a point unifies seemingly different segment relationships—and you'll handle any variation thrown at you. Don't just memorize facts; know what concept each theorem illustrates.

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Angle-Arc Relationships

These theorems establish the fundamental connection between angles and the arcs they intercept. The key principle: an angle's measure depends on its vertex position relative to the circle—at the center, on the circle, or outside it.

Central Angle Theorem

• A central angle equals its intercepted arc—this is the baseline from which all other angle-arc relationships derive
• Congruent central angles intercept congruent arcs, establishing a direct one-to-one correspondence
• Foundation theorem for circle measurement; the $360°$ in a circle comes from central angle reasoning

Inscribed Angle Theorem

• An inscribed angle measures half its intercepted arc—vertex on the circle "halves" what a central angle would measure
• Inscribed angles intercepting the same arc are congruent, regardless of where the vertex sits on the circle
• Proof strategy: draw the radius to create isoceles triangles and use the Exterior Angle Theorem

Thales' Theorem

• Any angle inscribed in a semicircle is a right angle—the diameter subtends a $180°$ arc, so the inscribed angle is $90°$
• Special case of the Inscribed Angle Theorem where the intercepted arc is exactly half the circle
• Construction tool: to create a right angle, draw a circle with your hypotenuse as diameter

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Tangent Properties

Tangent lines touch circles at exactly one point, creating unique perpendicularity and angle relationships. The defining property: a tangent is perpendicular to the radius at the point of tangency.

Tangent Line Theorem

• A tangent is perpendicular to the radius at the point of tangency—this $90°$ angle is the foundation of all tangent problems
• Two tangent segments from an external point are congruent, creating isoceles triangles useful in proofs
• Algebraic application: use the right angle to set up Pythagorean relationships between radius, tangent length, and distance to center

Tangent-Chord Theorem

• The angle between a tangent and chord equals half the intercepted arc—same as an inscribed angle would measure
• Think of it as a "degenerate" inscribed angle where one side becomes tangent to the circle
• Problem-solving tip: when you see a tangent meeting a chord, immediately look for the arc it cuts off

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Power of a Point

These theorems unify segment relationships through a single concept: the power of a point with respect to a circle is constant regardless of which line through that point you choose. For a point at distance $d$ from center of a circle with radius $r$, the power equals $d^2 - r^2$.

Chord-Chord Power Theorem

• When two chords intersect inside a circle: $AE \cdot EB = CE \cdot ED$ where $E$ is the intersection point
• Power is negative for interior points (both segments extend in opposite directions from the point)
• Algebraic setup: if one chord is bisected, you get a perfect square on one side of the equation

Tangent-Secant Theorem

• From an external point: $(\text{tangent})^2 = (\text{whole secant}) \times (\text{external segment})$
• Written algebraically: if tangent length is $t$ and secant has external part $a$ and internal part $b$, then $t^2 = a(a + b)$
• Special case: two secants from the same point give $a_1(a_1 + b_1) = a_2(a_2 + b_2)$

Intersecting Chords Theorem

• Products of chord segments are equal: $AE \cdot EB = CE \cdot ED$ for chords $AB$ and $CD$ intersecting at $E$
• Equivalent to Chord-Chord Power Theorem—different name, same relationship
• Proof approach: show triangles $\triangle AEC \sim \triangle DEB$ using inscribed angles on the same arc

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Chord and Diameter Relationships

These theorems describe how chords interact with each other and with the circle's diameter. Perpendicularity and bisection are the key themes here.

Perpendicular Chord Theorem

• A diameter perpendicular to a chord bisects that chord—and also bisects the arc
• Converse is also true: the perpendicular bisector of any chord passes through the center
• Construction use: to find a circle's center, draw perpendicular bisectors of two chords—they intersect at the center

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Cyclic Polygon Properties

When polygons are inscribed in circles, their angles satisfy special constraints. The inscribed angle theorem extends to create relationships between opposite angles.

Cyclic Quadrilateral Theorem

• Opposite angles of a cyclic quadrilateral are supplementary: $\alpha + \gamma = 180°$ and $\beta + \delta = 180°$
• Proof: opposite angles intercept arcs that together form the full circle ($360°$), so each pair of angles sums to $180°$
• Converse test: if a quadrilateral has supplementary opposite angles, it can be inscribed in a circle

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Quick Reference Table

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Self-Check Questions

1. Which two theorems both state that an angle equals half its intercepted arc, and what distinguishes where each angle's vertex is located?

2. A chord is 8 units long and sits 3 units from the center of a circle. Using the Perpendicular Chord Theorem, find the radius. What property lets you set up the right triangle?

3. Compare and contrast the Chord-Chord Power Theorem and the Tangent-Secant Theorem. How does the "power of a point" concept unify them?

4. If a quadrilateral has angles measuring $85°$, $95°$, $100°$, and $80°$, can it be inscribed in a circle? Which theorem justifies your answer?

5. FRQ-style: Given a circle with diameter $AC$, point $B$ on the circle, and a tangent at $B$ meeting line $AC$ extended at point $P$, identify which theorems you would use to find $\angle PBA$ and explain why each applies.