Given a line $AB$ and a point $C$ not on $AB$ it is easy enough to construct a perpendicular line to $AB$ passing through $C$ using two circles as demonstrated in the following picture.

perpendicular line construction

Here we pick two arbitrary points $E$ and $F$ on $AB$ and draw circles with centres $E$ and $F$ and radii $EC$ and $FC$. Then we take the two points of intersection of these circles (one of which is $C$) and draw a line between them. This gives a perpendicular line to $AB$ passing through $C$.

A similar proof is given in Euclid's Elements which uses the same idea and constructs two circles.

I am interested to know if we can construct a perpendicular from a given point to given line using the compass just a single time.

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    $\begingroup$ yes. Walk over to a guy who knows how and say you will stab him with the compass unless he constructs the perpendicular. $\endgroup$ – Will Jagy Jul 16 '17 at 17:54
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    $\begingroup$ @Will Jagy I appreciate. I would call this method "compass only construction under constraint." $\endgroup$ – Jean Marie Jul 16 '17 at 18:07
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    $\begingroup$ @Will, with that method you might even not use the compass! $\endgroup$ – Mariano Suárez-Álvarez Jul 16 '17 at 19:36
  • $\begingroup$ @Mariano, true. Note that Jack answered with no explicit threats. $\endgroup$ – Will Jagy Jul 16 '17 at 19:41
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    $\begingroup$ Well, he knows what is best for him... $\endgroup$ – Mariano Suárez-Álvarez Jul 16 '17 at 19:42

Yes we can! By exploiting the properties of orthocentric systems.

enter image description here

  1. Let $\Gamma$ be a small circle centered at $O\in\ell$, let $AB$ be its diameter on $\ell$;
  2. Let $C=PA\cap\Gamma$ and $D=PB\cap\Gamma$. Since $\widehat{ADB}=\widehat{ACB}=90^\circ$...
  3. By defining $E=AD\cap BC$ we have that $E$ is the orthocenter of $ABP$, hence...
  4. $PE\perp \ell$ as wanted.

Please don't stab me.

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    $\begingroup$ Beautiful. ${}$ $\endgroup$ – Mariano Suárez-Álvarez Jul 16 '17 at 19:38
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    $\begingroup$ @Jack D'Aurizio [+1] I was blocked by the fact that I wasn't on the track of a construction with a compass and a ruler. But anyhow, I wouldn't have thought to this clever construction ! $\endgroup$ – Jean Marie Jul 16 '17 at 19:42
  • $\begingroup$ @JeanMarie right, I thought if a straightedge were permitted it would have been mentioned. $\endgroup$ – Will Jagy Jul 16 '17 at 19:43
  • $\begingroup$ @WillJagy: I guessed the perpendicular had to be drawn somehow :D $\endgroup$ – Jack D'Aurizio Jul 16 '17 at 19:45
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    $\begingroup$ @WillJagy: there should be a bit of your humour in everybody's life :D $\endgroup$ – Jack D'Aurizio Jul 16 '17 at 20:07

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