Cyclic Quadrilaterals

Draw a circle. Now drop four dots anywhere on its edge and join them up, in order, into a four-sided shape. You have just made a cyclic quadrilateral — a quadrilateral whose four corners all sit exactly on one circle. (The word cyclic just means "on a circle".)

Here is the surprise. Measure the angle in one corner, then measure the angle in the corner directly opposite — across the shape. Add those two numbers together. You will get 180^\circ. Slide the dots around to make a completely different shape and try again: still 180^\circ. The circle silently ties every pair of opposite corners together so they always sum to a straight line.

\angle A + \angle C = 180^\circ \qquad \angle B + \angle D = 180^\circ

Angles that add to 180^\circ are called supplementary, so each pair of opposite angles in a cyclic quadrilateral is supplementary. That single fact turns a lot of scary-looking circle problems into a quick subtraction.

When all four vertices of a quadrilateral lie on a circle:

Why it works

Four points dropped onto a circle and joined up always behave the same way. Step through it and watch the two opposite angles.

Whatever the angle at A turns out to be — call it x — the angle directly across at C is exactly 180^\circ - x, so together they make 180^\circ. The same is true of the other pair, B and D.

Where does the 180^\circ come from? Each of the two opposite angles "stands on" one of the two arcs that the other two corners cut the circle into. By the angle at the centre theorem, each angle is half of its arc's centre angle. The two arcs together make the whole 360^\circ round the centre — so the two angles together are half of 360^\circ, which is 180^\circ.

Using the rule

Almost every cyclic-quadrilateral problem is really the same move: find the corner opposite the one you want, and subtract from 180^\circ.

Example 1 — just subtract

A cyclic quadrilateral ABCD has \angle A = 70^\circ and \angle B = 85^\circ. Find \angle C and \angle D.

\angle C is opposite \angle A, and \angle D is opposite \angle B:

\angle C = 180^\circ - 70^\circ = 110^\circ \qquad \angle D = 180^\circ - 85^\circ = 95^\circ

Quick check: all four add to 70^\circ + 85^\circ + 110^\circ + 95^\circ = 360^\circ, exactly as the four angles of any quadrilateral should.

Example 2 — a short angle-chase

Suppose a diameter is drawn as one of the diagonals, so one of the triangles inside is right-angled (that's the angle in a semicircle). If that gives you \angle B = 90^\circ, then the opposite angle is forced:

\angle D = 180^\circ - 90^\circ = 90^\circ

One circle theorem hands its answer straight to the next. Chasing angles like this — using each fact you find to unlock the following one — is the heart of circle geometry.

Example 3 — with algebra

Two opposite angles of a cyclic quadrilateral are (2x + 10)^\circ and (x + 20)^\circ. Find x.

Opposite angles are supplementary, so set their sum to 180^\circ and solve:

(2x + 10) + (x + 20) = 180 3x + 30 = 180 \;\Rightarrow\; 3x = 150 \;\Rightarrow\; x = 50

So the two angles are 110^\circ and 70^\circ — and, reassuringly, they add to 180^\circ.

The rule runs backwards too

The theorem also works in reverse — this is called the converse. If you meet a quadrilateral and find that a pair of its opposite angles already adds to 180^\circ, then its four corners must lie on a single circle: the shape is cyclic, and you could draw a circle neatly through all four corners.

That is genuinely useful: it is how you prove four points lie on a circle without ever drawing the circle — just check that a pair of opposite angles sums to 180^\circ.

The pair that sums to 180^\circ is the opposite angles — the two that face each other across the shape (A with C, B with D). It is not the adjacent (next-door, side-by-side) angles. Two angles that share a side have no special sum at all.

And there is a second trap: this rule only works when the shape really is cyclic — all four corners sitting on one circle. A general four-sided shape drawn on plain paper has no such relationship between any of its angles (only the grand total of all four is fixed, at 360^\circ). So before you subtract from 180^\circ, make sure you have been told, or can see, that the corners lie on a circle.

This is the four-cornered cousin of the angle in a semicircle: both come from the same angle-at-the-centre idea, one for a triangle on a diameter, one for a quadrilateral on a circle.

Cyclic quadrilaterals hide a deeper classical treasure called Ptolemy's theorem, which links the four sides of a cyclic quadrilateral to its two diagonals. Nearly two thousand years ago the astronomer Ptolemy used exactly this result to build a "table of chords" — an early version of the sine and cosine tables you meet in trigonometry. With those tables, ancient astronomers could calculate and predict where the planets would appear in the night sky, months in advance. A rule about corners on a circle turned out to be a key that helped unlock the heavens.