Circles and Angles, 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!