# Circles and Angles

## Some words

A **chord** of a circle is a line segment whose endpoints lie on the circle.

A **secant** of a circle is a line that intersects the circle at two points

An **inscribed angle** is the angle formed
by two chords having a common endpoint. The other endpoints define the
**intercepted arc**.

The **central angle** of the intercepted arc is the angle
at the midpoint of the circle.

In the picture to the left, the inscribed angle is the angle \(\angle ACB\),
and the central angle is the angle \(\angle AMB\).

## GeoGebra tasks

#### The central angle and the inscribed angle

What’s the relation between the central angle and the inscribed angle of an arc?

Make a conjecture and write it down. It must be clearly shown from your construction that your conjecture holds. Save your construction.

#### Inscribed angles

What can you say about different inscribed angles from the same arc? Can you figure it out without making a construction? Make a conjecture and write it down. Argue that your conjecture holds by either referring to a construction, or by reasoning (or both)

#### Opposite angles of a quadrilateral in a circle

Make a circle and place four points on the circle. Make a quadrilateral using the four points. What can you say about opposite angles of the quadrilaterals?

Make a conjecture and write it down. It must be clearly shown from your construction that your conjecture holds. Save your construction.

#### Inscribed angle in a semi-circle

What can you say about the inscribed angles in a semi-circle? Can you figure it out without making a construction? Make a conjecture and write it down. Argue that your conjecture holds by either referring to a construction, or by reasoning (or both).

#### Intersecting chords

Make a circle with two intersecting chords.

Make four variables to store distances as in the picture to the left. \(a=AE\), \(b=EC\), \(c=BE\) and \(d=ED\) (you can use other letters or names)

If you want to store the distance between the points \(A\) and \(B\) into a
variable called a, you write: `a=Distance[A,B]`

in the input bar. You can
also make a segment between the points \(A\) and \(B\), and use the value of the
segment.

Make sure you don't use variable-names already used by the program.

Find a relation between the four distances in terms of either ratios or products.

Make a conjecture and write it down. It must be clearly shown from your construction
that your conjecture holds. Save your construction.

## Proofs

When proving your conjectures, you are allowed to use following "facts"

#### The central angle and the inscribed angle

**Conjecture:** The central angle is twice as large as the inscribed
angle if they both are angles on the same intersecting arc.

**Proof:** There are three cases.

Try to prove the theorem in some of the case/cases. Case 3 is the most difficult case, you can use Case 1 and subtraction to prove Case 3.

#### Inscribed angles

**Conjecture:** Inscribed angles on the same
intersecting arc are all equal.

**Proof:** Use the theorem about inscribed and central angles!

#### Opposite angles of a quadrilateral in a circle

**Conjecture:** The sum of opposite angles of
a quadrilateral inscribed in a circle is 180°.

**Proof:** Take a closer look at the angles around the center
of the circle, and then use the theorem about inscribed and central angles!

#### Inscribed angle in a semi-circle, Thales' Theorem

**Conjecture:** The inscribed angle in a semi-circle
is 90°.

**Prove it!**

#### Intersecting chords

\(a=AE\), \(b=EC\), \(c=BE\) and \(d=ED\)

**Conjecture:** In the picture to the left,
\[ad=bc\Leftrightarrow \frac{a}{c}=\frac{b}{d}\]

**Proof:** Use the theorem about inscribed angles to find
similar triangles, and then prove it!

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License