Value of Position-Ratio

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