Joachimsthal's Section-Formulae

Theorem
Let $P = \tuple {x_1, y_1}$ and $Q = \tuple {x_2, y_2}$ be points in the Cartesian plane.

Let $R = \tuple {x, y}$ be a point on $PQ$ dividing $PQ$ in the ratio:


 * $PR : RQ = l : m$

Then:

Proof

 * Joachimsthals-section-formulae.png

Let the ordinates $PL$, $QM$ and $RN$ be constructed for $P$, $Q$ and $R$ respectively.

Then we have: