Circle Theorems

Some interesting things about angles and circles

Inscribed Angle

First off, a definition:

Inscribed Angle: an angle made from points sitting on the circle's circumference.

Inscribed angle ABC with vertex B on the circle circumference and endpoints A and C
A and C are "end points"
B is the "apex point"

Play with it here:

images/circle-prop.js?mode=inscribe

When you move point "B", what happens to the angle?

Inscribed Angle Theorems

Keeping the end points fixed ...

... the angle a° is always the same,
no matter where it is on the same arc between end points:

Inscribed angles with different vertices on the same arc showing they are all equal to a degrees
(Called the Angles Subtended by Same Arc Theorem)

And an inscribed angle a° is half of the central angle 2a°

Inscribed angle a on circumference and corresponding central angle 2a at the center
(Called the Angle at the Center Theorem)

Try it here (not always exact due to rounding):

images/circle-prop.js?mode=inscribe2

Example: What's the size of Angle POQ? (O is circle's center)

Circle with an inscribed angle PRQ measuring 62 degrees. O is the center of the circle, and central angle POQ needs to be found.

Angle POQ = 2 × Angle PRQ = 2 × 62° = 124°

Example: What's the size of Angle CBX?

Circle with points A, D, B, C, X. Angle ADB is 32 degrees, Angle BXC is 85 degrees. X lies on line segment BC.

Angle ADB = 32° also equals Angle ACB.

And Angle ACB also equals Angle XCB.

So in triangle BXC we know Angle BXC = 85°, and Angle XCB = 32°

Now use angles of a triangle add to 180° :

Angle CBX + Angle BXC + Angle XCB = 180°

Angle CBX + 85° + 32° = 180°

Angle CBX = 63°

Angle in a Semicircle (Thales' Theorem)

An angle inscribed across a circle's diameter is always a right angle:

Circle with a diameter and an inscribed angle subtended by this diameter, clearly marked as 90 degrees.
(The end points are either end of a circle's diameter,
the apex point can be anywhere on the circumference.)

Play with it here:

images/circle-prop.js?mode=thales

Why? Because:

The inscribed angle 90° is half of the central angle 180°

(Using "Angle at the Center Theorem" above)

Animation showing an inscribed angle of 90 degrees in a semicircle and its corresponding central angle of 180 degrees (a straight line).

Another Good Reason Why It Works

Animation showing an inscribed triangle in a semicircle being rotated around the center to form a rectangle, demonstrating the right angle.

Animation highlighting the sides and angles of the rectangle formed by rotating the inscribed triangle, emphasizing the 90-degree internal angles.

We could also rotate the shape around 180° to make a rectangle!

It is a rectangle, because all sides are parallel, and both diagonals are equal.

And so its internal angles are all right angles (90°).

Example: What's the size of Angle BAC?

Circle with a diameter and an inscribed triangle ABC, where angle ABC is 55 degrees. Angle BAC needs to be found.

The Angle in the Semicircle Theorem tells us that Angle ACB = 90°

Now use angles of a triangle add to 180° to find Angle BAC:

Angle BAC + 55° + 90° = 180°

Angle BAC = 35°

Animation showing an inscribed angle subtended by a diameter, with its vertex moving along the circumference, consistently maintaining 90 degrees.
So there we go! No matter where that angle is
on the circumference, it is always 90°

Finding a Circle's Center

Finding a circle's center by drawing two perpendicular diameters from right angles

We can use this idea to find a circle's center:

draw a right angle from anywhere on the circle's circumference, then draw the diameter where the two legs hit the circle
do that again but for a different diameter

Where the diameters cross is the center!

Drawing a Circle From 2 Opposite Points

When we know two opposite points on a circle we can draw that circle.

Put some pins or nails on those points and use a builder's square like this:

Drawing a circle by sliding a builder's square against two fixed pins

Because the corner of the square is always 90°, sliding it while keeping both sides touching the pins forces the corner to trace out a perfect semicircle.

Cyclic Quadrilateral

A "Cyclic" Quadrilateral has every vertex on a circle's circumference:
A Cyclic Quadrilateral's opposite angles add to 180°: a + c = 180° b + d = 180°

Example: What's the size of Angle WXY?

Cyclic quadrilateral WXYZ with angle WZY marked as 69 degrees. Angle WXY, opposite to WZY, needs to be found.

Opposite angles of a cyclic quadrilateral add to 180°

Angle WZY + Angle WXY = 180°

69° + Angle WXY = 180°

Angle WXY = 111°

A circle with a tangent line touching at a single point. A radius is drawn from the center to this point of tangency, forming a right angle (90 degrees) with the tangent line.

Tangent Angle

A tangent line just touches a circle at one point.

It always forms a right angle with the circle's radius.

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