Value of Position-Ratio
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Theorem
Let $P$ and $Q$ be points in space.
Let $R$ be a point on the straight line passing through $P$ and $Q$.
Let $k$ denote the position-ratio of $R$.
Then:
- $k = \dfrac {PQ} {RQ} - 1$
Proof
\(\ds k\) | \(=\) | \(\ds \dfrac {PR} {RQ}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {PQ + QR} {RQ}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {PQ} {RQ} - 1\) |
$\blacksquare$
Sources
- 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text I$. Coordinates: $11$. Position-ratio of a point