Investigation of Pedal Triangles Simson Line
By: Amanda Sawyer
Given a triangle and a point, we can create the pedal triangle by finding the perpendicular line from each of the three sides to the given point.† When the pedal triangleís three points are collinear, that line created by these three points is called the Simson Line.† During this investigation, we will see that this line does exist and it exist on the circumcircle.
Letís name the three points of our pedal triangle D, E, and F as seen above.† To create the Simson Line, we want to show that
This will be proven using the fact that PCAB is a cyclic quadrilateral.† That means that all four points must lie in a single circle.† This is easy to show because when we trace the point P to continue the Simson Line.† You can see this through the application PedalCircle.†
Since we have a cyclic quadrilateral, we know that
Next, we can now see that PEDB is also a cyclic quadrilateral by the picture below.††
With this information we can now use Thaleís Theorem to make generalization about two of the angles.† Remember Thaleís Theorem stated that if A, B and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle.† Therefore, we know that:
†Now, this shows us by the transitive property that
This property allows us to see that PFCE is also a cyclic quadrilateral.† Thus we know that
From this information, we can now conclude that:
From this proof, we are able to observe that we must have P as a point on our cirumcircle to create the properties of the cyclic quadrilateral that we needed to prove that the Simson Line exists at that point.