Simson line for olympiads

This is part of a series of articles based on my high school research paper from 2021 covering nonstandard approaches to solving olympiad problems. The problems from the Latvian Open Mathematics Olympiad mentioned in the article can be found in the LU NMS archive.

Let’s get to know a fairly rare, but certainly elegant theorem from geometry: the Simson line.

Simson line

The Simson line can be defined as follows:

Given altitudes drawn from a point $D$, which is on the circumcircle of $\triangle ABC$, to the lines $AB,BC,CA,$ the feet of the altitudes are on one line called the Simson line.

Below is an interactive diagram, where the point $D$ can be dragged around the circumcircle. The Simson line is drawn thickly and in red.

Olympiad example

Latvian Open Mathematics Olympiad 2015/2016, Form 12, Problem 4.
In triangle $ABC$, the angle bisector of $\angle ABC$ intersects the triangle’s circumcircle in point $D$. Segments $DP$ and $DQ$ are altitudes in triangles $ABD$ and $BCD$ respectively. Prove that the segment $PQ$ bisects $AC$!

A static diagram is given below.

From the given information, $PQ$ is definitely a Simson line. If $M$ is its intersection point with $AC$, then $DM\perp AC$. Now, if we assume that $M$ really is the midpoint of $AC$, that implies $DM$ is both an altitude and a median of $\triangle ADC$, which is possible if and only if $\triangle ADC$ is isosceles. So, we should first try to prove that $AD=DC$.

Since $AD$ is an angle bisector of $\angle ABC$, \[\angle ABD=\angle DBC\Longrightarrow\overparen{AD}=\overparen{DC}\Longrightarrow AD=DC,\] hence $\triangle ADC$ is isosceles.

The points $P,Q$ belong to a Simson line by definition, therefore, if we let the intersection point of $PQ$ and $AC$ be $M$, we have $DM\perp AC$. This means $DM$ ir an altitude of $\triangle ADC$, but because this triangle is isosceles, it is also a median and $AM=MC,$ which is what we had to prove.

Final thoughts

The Simson line appears infrequently in olympiads, though it is probably a good fit for very open selection contests, where the goal might be to see how deep each student’s theoretical knowledge is.

A good challenge is to try proving that the Simson line bisects the segment $DH$, where $H$ is the orthocenter of $\triangle ABC$. Best of luck!