1. What is a pedal triangle?
The pedal triangle DEF of a point P with respect to a triangle ABC is the triangle whose vertices are the feet of the perpendiculars dropped from P to the sides of triangle ABC (from http://www.daviddarling.info/encyclopedia/P/pedal_triangle.html)
Explore a pedal triangle using below file. (If you want GSP file, Click HERE!)
Move vertices of a given triangle and observe a pedal triangle when a given triangle is acute, obtuse, or right triangle.
How does a pedal triangle change according to the location of the point P?
2. What if the pedal point P is the centroid of triangle ABC?
In this file, the point P is located in the centroid of the triangle ABC.
3. What if the pedal point P is the incenter of triangle ABC?
Download GSP file(Click!) and then observe what kind of triangle is made.
As you can see, the result is the triangle formed by the points of tangency of the incircle of the triangle ABC, which is called 'contact triangle.'
Since the distances from the incenter to each side are equal,
.
By the construction of the pedal triangle,
.
Therefore, the triangle DEF becomes inscribed triangle of the incircle of the triangle ABC. In this case, the pedal point P becomes the circumcenter of the pedal triangle DEF.
4. What if the pedal point is the orthocenter of a triangle ABC?
Explore a shape of a pedal triangle when the pedal point is the orthocenter (In below file, the point P is an orthocenter of a triangle ABC).
If you want GSP file, Click HERE!
As you can see, the pedal triangle becomes the triangle DEF whose vertices are perpendicular feet from each vertex of the triangle ABC, which is called 'orthic triangle'.
Think about the case when the orthocenter is outside a given triangle, that is a given triangle is obtuse.
Even though the orthocenter is outside a triangle, the result is also an orthic triangle.
5. What if the pedal point P is the circumcenter of a given triangle ABC?
If you want GSP file,
Click HERE!
As you can observe using above file, the result is the triangle whose vertices are midpoints of sides of a given triangle ABC, which is called a medial triangle. Even though the pedal point P is outside a given triangle ABC, the result doesn't change.
Why? Since the circumcenter of a triangle is intersection of perpendicular bisectors, the points D, E, and F are midpoints of sides BC, AC, and AB, respectively and perpendicular feet from the point P.
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