[Math] Construct a perpendicular to a given line from a given (external) point, using a compass only once

euclidean-geometrygeometric-construction

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.

Best Answer

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

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  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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