Exploration
of the Simson Line

By: Sydney
Roberts

Let triangle ABC be any triangle, and P be any point in the plane. A triangle can then be constructed by taking the perpendiculars from P to each side of the triangle. These three intersection points can form the vertices of a new triangle, which we will call the Pedal Triangle. Refer below for an example, or for a GSP script tool click HERE.

Can this
triangle always be formed? For example, think of what would happen if P were at
one of the vertices. Without loss of generality, let’s assume the pedal point,
P, is at vertex A as pictured below.

Then we can
see that the pedal triangle become degenerate and forms a line. Why is this?
Well, remember that the pedal triangle’s vertices are constructed by the lines
that go through P and are perpendicular to the sides of the triangle. By choosing
P as A, then the intersection of a line that goes through P and is
perpendicular to AC is the point A. Also, the intersection of a line that goes
through P and is perpendicular to BA also becomes the point A. At this point,
two of our vertices for our pedal triangle lie at the same point. The third
vertex of the pedal triangle becomes one of the altitudes of triangle ABC since
the pedal point is at a vertex. Therefore, the three vertices of the pedal
triangle become collinear. This line that is formed is referred to as the *Simson** Line. *In order to see that the Simson Line is formed when P is at any vertex of the
triangle, use the script tool linked above to do your own investigations.

In order to
consider the other situations where the pedal triangle might form the Simson Line, think about what the location of the three
vertices of the original triangle ABC have in common. We know these three
vertices cause the pedal triangle to become degenerate, so could they possibly
give us a clue about the other condition? When thinking about this, I thought
about that you only need three points to define a circle. I also knew that
since all triangles are cyclic, that a circum circle exists that circumscribes
triangle ABC. Hence, all three vertices lie on the circumcircle. I then conjectured that if the pedal point
lies on the circumcircle, then the Simson Line would
be formed. This turned out to be true. Use the following GSP file to see this
for yourself. SIMSON LINE GSP FILE To
see that all points on the circumcircle form pedal points that will create the Simson Line, try animating the pedal point.

For a fun
extension, I tried tracing the Simson Line as the
pedal point moved around the circumcircle. I then noticed this interesting
shape form:

Click HERE for a GSP fil that will trace the Simson Line. Through
searching the internet, I discovered this shape is called the *Steiner deltoid* of the reference triangle.