Hard

Proving Four Points Are Concyclic

Simple Explanation

To formally prove four points are concyclic, identify a shared segment, compute (or use given information to find) the two relevant angles subtending it from the other two points, and show they are equal (or, for a quadrilateral, that opposite angles sum to 180°).

Why Do We Need It?

This combines the concyclic point conditions into a complete, exam-ready proof technique — showing not just recognizing the pattern, but constructing a rigorous argument.

Formula

A Condition for Concyclic Points

If ∠ACB = ∠ADB (C, D on the same side of AB), then A, B, C, D are concyclic

If two points, C and D, on the same side of a segment AB, both see AB under the exact same angle, then all four points A, B, C, D must lie on one common circle.

A, B: — two fixed points (forming the reference segment)
C, D: — two points on the same side of AB, each forming an angle with A and B

When to use it: Whenever you need to prove that four points all lie on a single circle, using only angle measurements.

Worked Example

Prove four points are concyclic using angle calculation

In quadrilateral WXYZ, ∠W = 110° and ∠Y = 70°. Prove W, X, Y, Z are concyclic.

Why Does This Work?

This uses the CONVERSE of the cyclic quadrilateral angle-sum theorem: if opposite angles of a quadrilateral sum to 180°, then a circle must pass through all four vertices — a fact provable by supposing a circle passes through three of the points and showing the fourth angle condition forces the last point onto that same circle too.

Real-Life Example

Verifying an engineering component fits a circular template

A quality-control engineer measures four corner angles of a manufactured part and needs to confirm it matches its intended circular design.

Checking the opposite-angle-sum condition (or an equal-angle condition) gives a fast confirmation the four measured points are consistent with lying on one true circle.

Practice

In quadrilateral PQRS, ∠P = 95° and ∠R = 84°. Are P, Q, R, S concyclic?

Hard

Common mistake

Accepting an angle sum that is "close to" 180° as sufficient proof — mathematical concyclic proofs require the condition to hold EXACTLY, not approximately.

Quick Review

For a quadrilateral: opposite angles summing to exactly 180° proves concyclic vertices.
For two points seeing a segment: equal angles (same side) proves concyclic.
Always check the condition holds exactly, not just approximately.