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Revising Circle Theorems
Going Round in Circles
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 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.
Overlapping lines sometimes cause confusion - the theorem still holds true...
The perpendicular bisector of a chord is a radius
The perpendicular bisector of a chord must pass through the centre of the circle. Not only is the chord bisected but the arc confined by the chord is bisected as well.
Popular in exam questions is the idea that radii from the centre of the circle to the ends of the chord will give congruent triangles. An isosceles triangle is also formed...
A radius and a tangent are perpendicular at the point they meet
Perpendicular lines always meet at right-angles (90°). A tangent is a line which only touches a circle's circumference at a single point. At any given point, there can only be one tangent and one perpendicular line from that point travelling through the centre of the circle (a radius).
This theorem leads to an interesting development...
Two tangents radiating from a single point outside the circle, C, will be equal in length at the ponts they touch the circle. Together with OA and OB, a kite is formed with ∠OAC and ∠OBC equal to 90°. A line from O to C will bisect the kite.
The angle between a chord and a tangent is equal to the angle in the alternate segment
Just like you would look for a cyclic quadrilateral, look for a 'cyclic triangle' in an exam question, with three points on the circumference and one meeting a tangent.
Another way of saying this is... where two chords meet on a tangent to the circle.
The angle subtended at the circumference in a semicircle is a right angle
Look for the diameter - it will always form the hypotenuse of the triangle. If you see this in an exam, there's usually something related to the 90° in the question.
Incidentally, this theorem is often attributed to Thales of Miletus hence it's sometimes known as Thales' Theorem.
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