Updated: Jul 1
Revising Circle Theorems
With a few straight lines and a circle, mathematicians have found patterns and developed theorems, along with their corresponding proofs, through logic and reasoning. Presented in no particular order, let’s see what they found out...
Angles at the circumference, subtended by the same arc, are equal
(Remember that an arc describes the section of the circumference between two points on the circumference).
'Subtended by' just means created or formed by an object.
Look out for 'hidden in plain sight' patterns. In the diagram above, the two triangles are also subtended from the ends of the same chord (in green) - some students see the chord and not the arc. In any case, the angles (x) at the circumference are the same.
Looked at another way, the equal angles are created by constructing chords from the ends of the same arc.
The sum of the opposite angles in a cyclic quadrilateral is 180°
Make sure that all four points (vertices) are touching the circumference of the circle to make use of this theorem.
For the geeks:
This circle is also known as the circumcircle and the vertices are said to be concyclic. Further, since the sides of the quadrilateral are all chords, a cyclic quadrilateral is also called a chordal quadrilateral.
The angle subtended by an arc at the centre is twice the angle subtended at the circumference by the same arc
In other words, the angle at the centre is double the angle at the circumference - or even... the angle at the circumference is half the angle at the centre. Or even... just kidding!
Overlapping lines sometimes cause confusion - the theorem still holds