Circle theorems
Maths · GCSE · The School
The words before the theorems
A chord joins two points on the circle. A tangent touches it once and never crosses. An arc is part of the edge; the region between a chord and its arc is a segment, and the wedge between two radii is a sector. A cyclic quadrilateral has all four corners on the circle. One more word does most of the work: an arc SUBTENDS an angle at a point when you draw a line from each end of the arc to that point. Almost every theorem below is a statement about the same arc subtending angles in two different places, so the first thing to find in any diagram is which arc is doing the work.
The four angle theorems
The angle at the centre is twice the angle at the circumference standing on the same arc. The angle in a semicircle is exactly 90 degrees, which is that first theorem with a straight line through the centre. Angles in the same segment are equal, so two angles standing on the same arc from the same side match. Opposite angles of a cyclic quadrilateral add to 180 degrees. Those four cover most of what an exam asks, and they all begin the same way: find the arc, then find what is standing on it.
The three tangent theorems
A tangent meets the radius at the point of contact at 90 degrees, which turns many problems into right-angled triangles. Two tangents drawn from the same outside point are equal in length, so the figure they make is symmetrical and gives you an isosceles triangle for free. The alternate segment theorem is the one people avoid: the angle between a tangent and a chord equals the angle in the alternate segment — the angle subtended by that chord from the far side of the circle.
A, B and C sit on a circle with centre O. Angle AOB at the centre is 130 degrees. Find angle ACB. Both stand on arc AB, and C is on the circumference, so the angle at the centre is twice the angle at the circumference: angle ACB is 65 degrees. Now suppose ABCD is a cyclic quadrilateral and angle ADC is 115 degrees. Opposite angles add to 180, so angle ABC is 65 degrees. Neither step needed measuring, and neither needed more than one theorem — most questions chain two or three short steps like these rather than hiding one hard one.
The reason is the mark
A circle-theorem question is marked on the chain of reasoning, not the number at the end. Write the theorem in words beside each line: "angle at centre is twice angle at circumference", "opposite angles in a cyclic quadrilateral sum to 180". A correct answer with no reasons loses most of the marks available, and a wrong answer with correct reasons keeps some. If a diagram looks like it needs a theorem you cannot name, mark on the radii and the right angles first and the shape usually declares itself.