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!