Fermat Problem

Theorem

Let $\triangle ABC$ be a triangle

Let the vertices of $\triangle ABC$ all have angles less than $120 \degrees$.

Let $\triangle ABG$, $\triangle BCE$ and $\triangle ACF$ be equilateral triangles constructed on the sides of $ABC$.

Let $AE$, $BF$ and $CG$ be constructed.

Let $P$ be the point at which $AE$, $BF$ and $CG$ meet.

Then $P$ is the Fermat-Torricelli point of $\triangle ABC$.

If one of vertices of $\triangle ABC$ be of $120 \degrees$ or more, then that vertex is itself the Fermat-Torricelli point of $\triangle ABC$.

Proof

The sum of the distances will be a minimum when the lines $PA$, $PB$ and $PC$ all meet at an angle of $120 \degrees$.

This is a consequence of the Fermat problem being a special case of the Steiner Tree Problem.

Consider the circles which circumscribe the $3$ equilateral triangles $\triangle ABG$, $\triangle BCE$ and $\triangle ACF$.

Consider quadrilaterals formed by $\triangle ABG$, $\triangle BCE$ and $\triangle ACF$ along with another point on each of those circumscribing circles.

Because these are cyclic quadrilaterals, the angle formed with these new points is $120 \degrees$.

It follows that $\Box APBG$, $\Box BPCE$ and $\Box APCF$ are those cyclic quadrilaterals.

Hence $\angle APC = \angle APB = \angle BPC = 120 \degrees$ and the result follows.

$\blacksquare$

Also known as

The Fermat Problem is also recognised as a special case of the Steiner Tree Problem for $3$ nodes.

Some refer to it as the Steiner Problem, or Steiner's Problem, for Jakob Steiner, but there are a number of such problems, and it is a good idea to be able to distinguish between them.

Some sources present this as Fermat's Problem.

Also see

• Definition:Extremum Problem, of which this is one of the first instances

Source of Name

This entry was named for Pierre de Fermat.

Historical Note

Pierre de Fermat sent the problem now known as the Fermat Problem to Evangelista Torricelli, who immediately solved it and returned the solution to Fermat straight away.

Hence the point in question bears the name of either or both mathematicians.

Sources

• 1973: Ross Honsberger: Mathematical Gems: Chapter $3$: Equilateral Triangles
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Fermat's problem or Steiner's problem
• 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Henry van Etten: $125$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Fermat point
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Fermat point