Rotation Theorem (Coni Method)

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.

The Coni method is not a separate theorem. It is a direct consequence of one fact about complex multiplication: multiplying by $e^{i\alpha}$ rotates a vector by $\alpha$ anticlockwise without changing its length.

The Core Fact

Let $w = re^{i\alpha}$. Multiplying any complex number $z$ by $w$:

$$wz = re^{i\alpha} \cdot |z|e^{i\theta} = r|z|\, e^{i(\theta + \alpha)}$$

The result has modulus $r|z|$ and argument $\theta + \alpha$. The angle increased by exactly $\alpha$. The length scaled by $r$.

When $r = 1$, the multiplication $e^{i\alpha} \cdot z$ is a pure rotation — same length, angle increased by $\alpha$.

Setting Up the Rotation

Let $z_1$, $z_2$, $z_3$ be three points in the complex plane. We want to describe the relationship: the segment $z_1 z_3$ is obtained by rotating segment $z_1 z_2$ about $z_1$ through angle $\alpha$ and scaling by factor $k$.

First, translate so $z_1$ is at the origin. The vectors from $z_1$ are:

$$\vec{v}_2 = z_2 - z_1, \qquad \vec{v}_3 = z_3 - z_1$$

The rotation-scaling that takes $\vec{v}_2$ to $\vec{v}_3$:

$$\vec{v}_3 = k e^{i\alpha} \cdot \vec{v}_2$$

where $k = |\vec{v}_3|/|\vec{v}_2|$ is the scaling factor and $\alpha$ is the rotation angle.

Substituting back:

$$z_3 - z_1 = k e^{i\alpha} (z_2 - z_1)$$

The Coni Formula

Dividing both sides by $(z_2 - z_1)$:

$$\frac{z_3 - z_1}{z_2 - z_1} = k e^{i\alpha} = \frac{|z_3 - z_1|}{|z_2 - z_1|}\, e^{i\alpha}$$

This single equation carries two pieces of information simultaneously:

• Its modulus gives $k = |z_3 - z_1|/|z_2 - z_1|$ — the ratio of the segment lengths.
• Its argument gives $\alpha$ — the angle from $z_1 z_2$ to $z_1 z_3$, measured anticlockwise.

The Isoceles Case

When the triangle $z_1 z_2 z_3$ is isoceles with $|z_3 - z_1| = |z_2 - z_1|$, the modulus of the ratio is 1, and the formula simplifies to:

$$\frac{z_3 - z_1}{z_2 - z_1} = e^{i\alpha}$$

This is the form most JEE problems use. Given $z_1$, $z_2$, and the angle $\alpha$, the third vertex $z_3$ of an isoceles (or equilateral when $\alpha = \pm\pi/3$) triangle is:

$$z_3 = z_1 + (z_2 - z_1)\,e^{i\alpha}$$

Why the Formula Is Written as a Ratio

The ratio $(z_3 - z_1)/(z_2 - z_1)$ might seem like an unusual object. But it is natural: it is the complex number that, when multiplied by the vector $z_1 z_2$, produces the vector $z_1 z_3$. Its argument is the angle between the two segments; its modulus is the ratio of their lengths. One complex number encodes both the geometry of angle and the geometry of distance at once.