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Formulas/maths/M4a Geometry/Condition for Four Concyclic Points

Condition for Four Concyclic Points

Four points z₁, z₂, z₃, z₄ lie on a common circle (or line) iff their cross-ratio is real. This is a deep result — it says the cross-ratio is invariant under Möbius transformations. Equivalently, arg((z₁−z₃)(z₂−z₄)/((z₁−z₄)(z₂−z₃))) = 0 or π.
Derivation

Four points lie on a common circle (or line) iff a specific ratio formed from them is real. This is the complex form of a classical circle theorem.

The Inscribed Angle Connection

Consider four points z1,z2,z3,z4z_1, z_2, z_3, z_4. Look at the angles:

α=arg⁡ ⁣(z3−z1z3−z2)(angle at z3 subtended by chord z1z2)\alpha = \arg\!\left(\frac{z_3 - z_1}{z_3 - z_2}\right) \quad \text{(angle at } z_3 \text{ subtended by chord } z_1 z_2\text{)} β=arg⁡ ⁣(z4−z1z4−z2)(angle at z4 subtended by chord z1z2)\beta = \arg\!\left(\frac{z_4 - z_1}{z_4 - z_2}\right) \quad \text{(angle at } z_4 \text{ subtended by chord } z_1 z_2\text{)}

Inscribed angle theorem: If z3z_3 and z4z_4 lie on the same arc (same side of chord z1z2z_1 z_2), these angles are equal. If on opposite arcs, they differ by π\pi.

In both cases, α−β=0\alpha - \beta = 0 or π\pi, meaning:

arg⁡ ⁣(z3−z1z3−z2÷z4−z1z4−z2)=0 or π\arg\!\left(\frac{z_3 - z_1}{z_3 - z_2} \div \frac{z_4 - z_1}{z_4 - z_2}\right) = 0 \text{ or } \pi

The Cross-Ratio

The expression inside the argument is the cross-ratio:

[z1,z2;z3,z4]=(z3−z1)(z4−z2)(z3−z2)(z4−z1)⋅(z4−z1)(z4−z1)[z_1, z_2; z_3, z_4] = \frac{(z_3 - z_1)(z_4 - z_2)}{(z_3 - z_2)(z_4 - z_1)} \cdot \frac{(z_4 - z_1)}{(z_4 - z_1)}

More cleanly, the equivalent form:

(z1−z3)(z2−z4)(z1−z4)(z2−z3)\frac{(z_1 - z_3)(z_2 - z_4)}{(z_1 - z_4)(z_2 - z_3)}

The argument of this expression is 00 or π\pi iff it is real. Therefore:

z1,z2,z3,z4 concyclic  ⟺  (z1−z3)(z2−z4)(z1−z4)(z2−z3)∈Rz_1, z_2, z_3, z_4 \text{ concyclic} \iff \frac{(z_1 - z_3)(z_2 - z_4)}{(z_1 - z_4)(z_2 - z_3)} \in \mathbb{R}

Why Lines Are Included

A line is a circle of infinite radius. When the four points are collinear, the inscribed angle theorem still holds in the limiting sense — the angles α\alpha and β\beta are both 00 or π\pi (the points all subtend 0°0° or 180°180°). The cross-ratio condition captures this automatically.

Practical Check

To verify four given points are concyclic: compute the cross-ratio and check that Im⁡(cross-ratio)=0\operatorname{Im}(\text{cross-ratio}) = 0. If the imaginary part is zero, they lie on a common circle or line.

In JEE problems, concyclicity often appears as a condition to be proved rather than computed — substituting the cross-ratio form and showing it is real is the standard approach.