Complex Numbers & Coordinates in Geometry

When the synthetic route stalls, compute.

Place the circumcircle as the unit circle in $\mathbb{C}$. Then $|a|=|b|=|c|=1$, conjugation is $\bar a = 1/a$, and the classical centres become formulas: the orthocentre is simply $h = a+b+c$.

Rotation by $\theta$ about the origin is multiplication by $e^{i\theta}$, which makes spiral similarity almost trivial to write down.

Coordinates are the blunter cousin: choose axes so the most points get zeros. Neither is elegant, but on a timed paper a guaranteed computation beats an elusive synthetic insight.

More techniques

• The Ravi SubstitutionKill the triangle inequality by construction.
• Phantom PointsDefine the point you want, then prove it is the one you have.
• All 26 techniques →