Geometry of Complex Numbers

Every complex number z = a+ib corresponds to the point P(a, b) in the Argand plane and to the position vector OP. The modulus |z| is the length of OP. This correspondence is the foundation of all geometric interpretations that follow.

|z₁−z₂| is simultaneously the modulus of the complex number (z₁−z₂) and the Euclidean distance between the points z₁ and z₂. This single expression replaces √((x₁−x₂)²+(y₁−y₂)²) throughout complex geometry.

Adding z₁ and z₂ corresponds to the parallelogram law of vector addition: z₁+z₂ is the diagonal of the parallelogram formed by z₁ and z₂. Subtraction z₁−z₂ gives the other diagonal direction, from z₂ to z₁.

Multiplying z by w = re^(iα) rotates z by angle α anticlockwise and scales its modulus by r. Multiplying by i (where r=1, α=π/2) is a pure 90° anticlockwise rotation. This geometric view of multiplication is the key to understanding the Coni method.

The complex number dividing the segment joining z₁ and z₂ internally in the ratio m:n. Identical in structure to the Cartesian section formula, with complex numbers replacing coordinates directly.

External division in ratio m:n (m ≠ n). Midpoint is the special case m = n = 1 of internal division. The midpoint formula z = (z₁+z₂)/2 is used constantly in complex geometry — it is simpler than its Cartesian form.

For triangle with vertices z₁, z₂, z₃, where a = |z₂−z₃|, b = |z₃−z₁|, c = |z₁−z₂| are the side lengths opposite to the respective vertices. The circumcentre has no simpler form than the Cartesian version, but the centroid and incentre are elegant.

α is the angle through which the segment z₁z₂ must be rotated anticlockwise about z₁ to reach the segment z₁z₃. If the triangle z₁z₂z₃ is isoceles with |z₃−z₁| = |z₂−z₁|, the ratio simplifies to e^(iα). This is the single most powerful tool in complex geometry.

Rotating the point z by angle α anticlockwise about the origin gives z' = ze^(iα). Rotation by 90°: z' = iz. Rotation by 180°: z' = −z. Rotation by −90°: z' = −iz. This is the Coni method with z₁ = 0.

Rotating z by angle α about the point z₀: shift z₀ to origin, rotate, shift back. Rearranged: z' = z₀ + (z−z₀)e^(iα). This is the general form of the Coni method and is used to find the third vertex of an equilateral triangle, rotate a line, or construct geometric figures.

Three points are collinear iff the ratio (z₃−z₁)/(z₂−z₁) is real — equivalently, arg((z₃−z₁)/(z₂−z₁)) = 0 or π. The determinant form: the 3×3 determinant with rows (zₖ, z̄ₖ, 1) equals zero. Both forms are used in JEE problems.

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.

To shift the origin to z₀, replace every z with Z + z₀. In the new coordinate system, z₀ maps to 0. This simplifies loci and equations centred at z₀ — e.g., a circle |z−z₀| = r becomes |Z| = r. Identical to the Cartesian origin-shift but in a single complex line.

z and z* are inverse points w.r.t. the circle |z−z₀| = r if they lie on the same ray from z₀ and the product of their distances from z₀ equals r². Equivalently: (z−z₀)(z̄*−z̄₀) = r². The inverse of a point inside the circle lies outside, and vice versa.