Circle Language Lets You Describe Geometry Precisely
- Circles appear simple, but a lot of powerful geometry comes from naming their parts accurately and using a small set of reliable facts (the circle theorems).
- In this section you will review the key parts of a circle and then build the essential angle and tangent properties used in problem solving.
Circumference
The curved boundary (perimeter) of a circle.
Center
The point inside a circle that is the same distance from every point on the circumference.
Radius
A line segment from the center of a circle to a point on the circumference.
Diameter
A chord that passes through the center of the circle, its length is twice the radius.
Chord
A line segment joining two points on the circumference of a circle.
Arc
A part of the circumference of a circle between two points.
Segment
The region bounded by a chord and the arc between the chord’s endpoints.
Secant (in the circle)
A line that intersects a circle at two points.
Tangent (to a circle )
A straight line that touches the circle's circumference at exactly one point, called the point of tangency, without crossing into the interior
- A line segment connects two endpoints and has a fixed length, but a line extends infinitely in both directions.
- This matters when deciding whether something is a chord (segment) or a secant (line).
Tangents Touch At One Point And Create A Right Angle With The Radius
A tangent to a circle touches the circle at exactly one point (the point of contact).
A key property links tangents to the circle's center:
Tangent–radius theorem: If a line is tangent to a circle at point $T$ and $OT$ is the radius to the point of contact, then $$OT \perp \text{tangent at }T$$
In other words, the tangent is perpendicular to the radius drawn to the contact point.
- Think of the radius as a "spoke" pointing straight out to the rim.
- At the rim, the tangent is like the direction a wheel would roll at that exact point, it is at right angles to the spoke.
- Do not assume a line is tangent just because it "looks like" it touches the circle.
- In proofs you usually need a reason, for example, "it meets the radius at $90^\circ$" or "it meets the circle at only one point."
Angles At The Center Are Twice Angles At The Circumference
Many circle problems involve angles subtended by the same chord or arc.
Angle subtended
An angle formed when two lines from a point meet two points on a circle. For example, chord $AC$ subtends an angle at the center $\angle AOC$ and an angle at the circumference $\angle ABC$.
Central angle theorem: For points $A,B,C$ on a circle with center $O$ (and with $B$ on the same side of chord $AC$ as the arc being used), $$\angle AOC = 2\angle ABC$$
So, the angle at the center is double the angle at the circumference standing on the same arc $AC$.
From $\angle AOC = 2\angle ABC$, you can write $$\angle ABC = \tfrac12\angle AOC$$
This is especially useful when you can find or relate a central angle easily (because radii form isosceles triangles).
- When a diagram includes the center $O$, look for triangles like $\triangle AOC$ where $OA=OC$ (radii).
- Isosceles triangles create equal base angles, which often unlock the whole proof.
The Angle In A Semicircle Is A Right Angle
If $AC$ is a diameter, then any point $B$ on the circle gives a right angle at $B$.
Angle in a semicircle theorem: If $AC$ is a diameter, then $$\angle ABC = 90^\circ$$
This follows from the central angle theorem: the angle subtended by diameter $AC$ at the center is $\angle AOC=180^\circ$, so the angle at the circumference is half of that.
If you know a triangle is drawn in a circle and one side is a diameter, you can immediately mark the opposite angle as $90^\circ$, then use right-triangle methods (Pythagoras or trigonometry) if lengths are involved.
Angles In The Same Segment Are Equal
If two angles stand on the same chord (or the same arc), they are equal.
Same segment theorem: If $A,B,C,D$ lie on the circumference and chord $AC$ is fixed, then $$\angle ABC = \angle ADC$$ provided both angles subtend chord $AC$ from the same side.
This is an immediate consequence of the central angle theorem: both angles are half of the same central angle $\angle AOC$.
- The phrase "same segment" means the angles must be on the same side of the chord.
- If one point is on the opposite side, the angles add to $180^\circ$ instead of being equal.
Opposite Angles In A Cyclic Quadrilateral Sum To 180°
A cyclic quadrilateral is a four-sided shape whose vertices all lie on a circle.
Cyclic quadrilateral
A quadrilateral whose vertices all lie on the circumference of the same circle.
Cyclic quadrilateral theorem: If $A,B,C,D$ lie on a circle, then $$\angle ABC + \angle ADC = 180^\circ,$$ and similarly the other opposite pair also sum to $180^\circ$.
The idea is that these two angles stand on complementary arcs whose central angles add to $360^\circ$, so the corresponding circumference angles add to $180^\circ$.
- In angle-chasing questions, look for any four points on the same circle.
- The moment you can justify "cyclic," you can convert a hard angle into "$180^\circ$ minus something."
Tangent–Chord Angles Link Circle Geometry To Straight Lines
A powerful tangent fact connects an angle made with a tangent to an angle in the circle.
Alternate segment theorem: If a tangent touches the circle at $A$ and chord $AB$ is drawn, then the angle between the tangent and chord $AB$ equals the angle in the opposite segment, for example $$\angle(\text{tangent at }A,\ AB)=\angle ACB,$$ where $C$ is a point on the circle on the opposite side of chord $AB$.
This theorem is frequently combined with parallel lines to prove triangles are isosceles or to show two triangles are similar.
- Suppose $ABC$ is tangent at $B$ and a chord $BD$ is drawn.
- The angle between the tangent $BC$ and chord $BD$ can be replaced by an angle at the circumference standing on chord $BD$.
- This substitution often turns a "line-and-circle" problem into a pure circle-angle problem.
Problem-Solving Patterns From Typical Proof Questions
Many circle-theorem proofs (including those like the practice problems in your course) rely on the same small set of moves.
Pattern 1: Convert An Angle Between Tangents Into A Central Angle
- If tangents at $X$ and $Y$ meet at $Q$, then radii $OX$ and $OY$ are perpendicular to the tangents, so quadrilateral $OXQY$ has two right angles.
- This gives a relationship between $\angle XQY$ and the central angle $\angle XOY$, and then you can use "center is twice circumference" to relate to an angle like $\angle XPY$.
- Whenever two tangents meet, draw the radii to the points of contact and mark the two right angles.
- That almost always creates a quadrilateral where angle sums become useful.
Pattern 2: Use Cyclic Quadrilaterals To Prove Similarity
- If two lines are extended to meet outside a circle (for example, chords extended to meet at $X$), angle equality often comes from:
- angles in the same segment, and
- vertically opposite angles at the external intersection.
- Once you have two equal angles, triangle similarity usually follows.
Pattern 3: Prove Isosceles By Showing Base Angles Are Equal
Triangles inside circle problems become isosceles when you can prove two angles are equal. Common reasons:
- radii create isosceles triangles (two radii are equal),
- same-segment angles are equal, or
- alternate segment theorem converts a tangent angle into a circumference angle.
Converse Statements Matter In Geometry Proofs
- Many theorems are written in the form "If $p$, then $q$."
- The converse swaps them: "If $q$, then $p$." For example:
- Theorem: If $AC$ is a diameter, then $\angle ABC=90^\circ$.
- Converse: If $\angle ABC=90^\circ$, then $AC$ is a diameter.
- In circle geometry, converses are often true and are extremely useful for proving points lie on a circle (or that a line is tangent).
- The converse of a true statement is not automatically true.
- In geometry you must justify it (either by a known converse theorem or a separate proof).
- A line touches a circle at $T$ and you draw radius $OT$. What angle do you immediately know?
- You see a quadrilateral with all vertices on the same circle. What is the key angle-sum fact?
- Two angles stand on the same chord $AC$ on the same side. How are they related?
- A triangle is inscribed in a circle and one side is a diameter. What is the opposite angle?
