# 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 `ACB`,
and the central angle is the 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 centre
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,
\[ab=cd\Leftrightarrow \frac{a}{c}=\frac{d}{b}\]

**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