Circle Theorems

Why This Matters

Circle theorems form the backbone of geometric reasoning about one of math's most fundamental shapes. They test your ability to recognize angle-arc relationships, power of a point, and perpendicularity conditions. These theorems connect algebraic manipulation with geometric intuition, and they show up repeatedly in proofs, coordinate geometry problems, and optimization questions.

What makes circle theorems powerful is 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, like why an inscribed angle is half its arc and how the power of a point unifies seemingly different segment relationships, and you'll handle any variation thrown at you.

---

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 where its vertex sits relative to the circle (at the center, on the circle, inside, or outside).

Central Angle Theorem

A central angle has its vertex at the center of the circle, and its measure equals its intercepted arc exactly. This is the baseline from which every other angle-arc relationship derives.

• Congruent central angles intercept congruent arcs (and congruent chords), giving you a direct one-to-one correspondence
• The fact that a full rotation measures $360°$ is what anchors central angle reasoning

Inscribed Angle Theorem

An inscribed angle has its vertex on the circle, and it measures exactly half its intercepted arc. The vertex being on the circle rather than at the center is what "halves" the angle.

• All inscribed angles that intercept the same arc are congruent, no matter where the vertex sits on the major arc
• Proof strategy: Draw the radius to the vertex to create isosceles triangles (two sides are radii), then use the fact that an exterior angle of a triangle equals the sum of the two remote interior angles

Thales' Theorem

Any angle inscribed in a semicircle is a right angle. A diameter subtends a $180°$ arc, so the inscribed angle is $\frac{180°}{2} = 90°$.

This is a special case of the Inscribed Angle Theorem where the intercepted arc is exactly half the circle. It's also a handy construction tool: to create a right angle, draw a circle with your hypotenuse as diameter, and any point on the circle (other than the endpoints) completes a right triangle.

Angles Inside and Outside the Circle

Two more angle-arc relationships round out the picture, depending on where the vertex falls:

• Vertex inside the circle (two chords crossing): The angle equals half the sum of the two intercepted arcs. If arcs $a$ and $b$ are intercepted, the angle is $\frac{a + b}{2}$.
• Vertex outside the circle (two secants, two tangents, or a secant and a tangent): The angle equals half the difference of the two intercepted arcs: $\frac{a - b}{2}$, where $a$ is the larger arc.

These follow logically from the Inscribed Angle Theorem. Moving the vertex inside the circle "adds" arc contributions; moving it outside "subtracts" them.

---

Tangent Properties

A tangent line touches a circle at exactly one point, creating unique perpendicularity and angle relationships. The defining property: a tangent is perpendicular to the radius drawn to the point of tangency.

Tangent Line Theorem

The tangent is perpendicular to the radius at the point of tangency, giving you a guaranteed $90°$ angle. This right angle is the foundation of nearly every tangent problem.

• Two tangent segments from the same external point are congruent. This creates an isosceles configuration (and actually forms a kite with the two radii), which is useful in proofs and for finding unknown lengths.
• Because of the right angle, you can set up Pythagorean Theorem relationships between the radius, the tangent length, and the distance from the external point to the center.

For example, if a circle has radius 5 and an external point is 13 units from the center, the tangent segment has length $\sqrt{13^2 - 5^2} = \sqrt{144} = 12$.

Tangent-Chord Theorem

The angle formed between a tangent and a chord drawn from the point of tangency equals half the intercepted arc. This gives the same "half the arc" result as an inscribed angle.

You can think of this as a limiting case of an inscribed angle: imagine sliding the vertex of an inscribed angle along the circle until it reaches the point of tangency. The angle approaches the tangent-chord angle, and the half-arc relationship still holds.

When you see a tangent meeting a chord, immediately identify the arc that the chord cuts off on each side. The angle between the tangent and the chord equals half the arc on the side of the angle.

---

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 the center of a circle with radius $r$, the power equals $d^2 - r^2$.

• If the point is inside the circle, the power is negative ($d < r$).
• If the point is on the circle, the power is zero.
• If the point is outside the circle, the power is positive ($d > r$).

The sign doesn't usually matter for solving problems, but understanding it helps you see why all three segment theorems below are really the same idea.

Chord-Chord Power Theorem (Intersecting Chords)

When two chords intersect at a point $E$ inside a circle, the products of their segments are equal:

$AE \cdot EB = CE \cdot ED$

where $AB$ and $CD$ are the two chords.

• The proof relies on showing that $\triangle AEC \sim \triangle DEB$ using inscribed angles that intercept the same arc, then writing the proportion of corresponding sides.
• If one chord is a diameter, or if one chord happens to be bisected at $E$, you get a perfect square on one side of the equation, which simplifies the algebra.

Note: You may see this listed separately as the "Intersecting Chords Theorem." Same relationship, different name.

Secant-Secant and Tangent-Secant Theorems

From an external point, the power of a point gives you two related results:

• Two secants: $a_1(a_1 + b_1) = a_2(a_2 + b_2)$, where $a$ is the external segment and $b$ is the internal chord portion of each secant. In other words, you multiply each secant's external segment by its whole length.
• Tangent and secant: $t^2 = a(a + b)$, where $t$ is the tangent length, $a$ is the external part of the secant, and $b$ is the part inside the circle.

The tangent-secant formula is really just the secant-secant formula where one secant has its two intersection points collapsed into a single tangent point (so $b = 0$ and the "whole secant" equals the "external segment," giving $t^2$).

---

Chord and Diameter Relationships

These theorems describe how chords interact with the diameter. Perpendicularity and bisection are the recurring themes.

Perpendicular Chord Theorem

A diameter that is perpendicular to a chord bisects that chord and also bisects both arcs created by the chord.

The converse is equally useful: the perpendicular bisector of any chord passes through the center of the circle. This gives you a practical construction method for finding a circle's center:

1. Draw any two non-parallel chords.
2. Construct the perpendicular bisector of each chord.
3. The point where the two perpendicular bisectors intersect is the center.

Another useful fact: congruent chords are equidistant from the center, and chords equidistant from the center are congruent. Distance here means the perpendicular distance from the center to the chord.

---

Cyclic Polygon Properties

When a polygon is inscribed in a circle (all vertices on the circle), its angles satisfy special constraints that follow directly from the Inscribed Angle Theorem.

Cyclic Quadrilateral Theorem

Opposite angles of a cyclic quadrilateral are supplementary:

$\alpha + \gamma = 180° \quad \text{and} \quad \beta + \delta = 180°$

The proof is straightforward: opposite angles intercept arcs that together make up the full circle ($360°$). Since each inscribed angle is half its intercepted arc, the two opposite angles sum to $\frac{360°}{2} = 180°$.

The converse is also true and serves as a test: if a quadrilateral has supplementary opposite angles, then it can be inscribed in a circle (it is cyclic).

---

Quick Reference Table

---

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.