Write-Up #9
Pedal Triangles
By Jaepil Han
2a. What if pedal point P is the centroid of triangle ABC?
Here's the graph of the pedal triangle and the centroid of triangle ABC.
Now, consider the case when the pedal point P is the centroid of triangle ABC.
Since PT, PR, and PS, are the perpendicular segments from the pedal point P to each side of triangle ABC, the sum of the areas of the triangles, ABP, BCP, and CAP, is the same as the area of triangle ABC.
Here's the proof that the six small triangles have the same areas.
Thus, the below follows.
Now, consider the pedal point P as the centroid of triangle ABC, and use the properties of the centroid of a triangle.
Since the centroid G of a triangle is the point of concurrency of the three medians, M1, M2, and M3 are the mipoints of the segments, AB, BC, and CA.
2b. What if . . . P is the incenter . . . ?
Now, consider the case when the pedal point is the incenter of triangle ABC.
After this point, we may use the properties that come from the pedal point P is the incenter of triangle ABC.
2c. What if . . . P is the Orthocenter . . . ? Prove that the Pedal triangle is the Orthic triangle of the original triangle. Even if the Orthocenter outside ABC?
Consider the case when the pedal point P is the orthocenter H of triangle ABC. Here's the graph when the pedal point P is the orthocenter H of triangle ABC.
Since the pedal point P is the orthocenter of triangle ABC, the points, R, S, and T, are the intersections of the lines which include the altitudes of triangle ABC and each side.
Therefore, the pedal triangle RST is the orthic triangle RST of triangle ABC by definition of orthic triangle.
2d. What if . . . P is the Circumcenter . . . ? Prove that the Pedal Triangle is the Medial Triangle of the original triangle. Even if the circumcenter is outside ABC?
Using the script tools, graph the pedal point P and the circumcenter C of triangle ABC.
Now, consider the case when the pedal point P is the circumcenter of triangle ABC.
Since the circumcenter C of triangle ABC is the point of concurrency of the perpendicular bisectors of each side of triangle ABC, the segments PR, PS, and PT are on on the perpendicular bisectors of each side of triangle ABC.
By definition of circumcenter and circumcircle, the three points, S, T, and R are the mipoints of each side also. Say the mipoints are M1, M2, and M3.
Therefore, by definition of medial triangle, the pedal triangle RST is the medial triangle of triangle ABC.
