Joachimsthal's Section-Formulae
Jump to navigation
Jump to search
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:
| \(\ds x\) | \(=\) | \(\ds \dfrac {l x_2 + m x_1} {l + m}\) | ||||||||||||
| \(\ds y\) | \(=\) | \(\ds \dfrac {l y_2 + m y_1} {l + m}\) |
Proof
Let the ordinates $PL$, $QM$ and $RN$ be constructed for $P$, $Q$ and $R$ respectively.
Then we have:
| \(\ds OL\) | \(=\) | \(\ds x_1\) | ||||||||||||
| \(\ds OM\) | \(=\) | \(\ds x_2\) | ||||||||||||
| \(\ds ON\) | \(=\) | \(\ds x\) | ||||||||||||
| \(\ds LP\) | \(=\) | \(\ds y_1\) | ||||||||||||
| \(\ds MQ\) | \(=\) | \(\ds y_2\) | ||||||||||||
| \(\ds NR\) | \(=\) | \(\ds y\) | ||||||||||||
| \(\ds LN : NM = PR : RQ\) | \(=\) | \(\ds l : m\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac {x - x_1} {x_2 - x}\) | \(=\) | \(\ds \dfrac l m\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \dfrac {l x_2 + m x_1} {l + m}\) | |||||||||||
| \(\ds y\) | \(=\) | \(\ds \dfrac {l y_2 + m y_1} {l + m}\) |
$\blacksquare$
Source of Name
This entry was named for Ferdinand Joachimsthal.
Historical Note
This result is attributed to Ferdinand Joachimsthal by D.M.Y. Sommerville in his $1924$ work Analytical Conics.
Further reference to Joachimsthal's Section-Formulae cannot be found online, although some considerably more sophisticated results which bear similar names are apparent.
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text I$. Coordinates: $10$. Joachimsthal's section-formulae