Four-Line Locus

Theorem

Let $4$ straight lines in the plane be given.

The problem is to find the locus of a point $P$ such that the product of the distances from $P$ to any $2$ of them is proportional to the product of the distances to the other $2$.


Solution

The required locus is a conic section.


Proof


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Historical Note

The problem of the was raised by Apollonius of Perga.


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Problem of Pappus
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): problem of Pappus
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