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, letís 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 triangleís 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.