Triangle Inequality: |z1 + z2| <= |z1| + |z2|

• Equality: arg(z1) = arg(z2) (same direction)

Reverse Triangle Inequality: ||z1| - |z2|| <= |z1 + z2|

• Equality: arg(z1) - arg(z2) = pi (opposite directions)

Combining both: ||z1| - |z2|| <= |z1 + z2| <= |z1| + |z2|

Max/Min of |z| given |z - z0| = r: Circle centered at z0 with radius r.

• max|z| = |z0| + r
• min|z| = |z0| - r if |z0| > r (origin outside circle)
• min|z| = 0 if |z0| <= r (origin inside or on circle)

Max/Min of |z - a| given |z - b| = r:

• max|z-a| = |a-b| + r
• min|z-a| = ||a-b| - r|

Sum of distances: min(|z-z1| + |z-z2|) = |z1-z2| (achieved on segment z1z2)

JEE Strategy for Max/Min problems:

1. Identify the geometric constraint (circle, disk, line)
2. Identify what to maximize/minimize (distance from some point)
3. Draw the Argand plane picture
4. Use geometric reasoning: farthest/nearest point on circle from given point