by Oktay Mercimek
I would like to discuss the situation that Pedal point p is the Circumcenter of the triangle.
Let's draw a triangle, its circumcenter and circumcircle using the tool that we made at Assignment 5.
We can call this point P because we think it as Pedal Point. and draw pedal triangle.
Question is "is this pedal triangle have another properties?"
We know circumcenter is equal distance from three vertexes.
We know Triangle APB is a isosceles triangle and PE is perpendicular to AB.
Then it requires |AE|=|EB|. (1)
We can apply same same logic to Triangle APC and Triangle BCP
then |AG|=|GC| (2) and |BF|=|FC| (3)
When we put together (1), (2) and (3), it means E, F and G are midpoints.
In this case Triangle EGF is medial triangle and pedal triangle. Click HERE to open GSP file of picture above
Let's look what happens when Triangle ABC is a obtuse triangle.
In this case P (or circumcenter) is outside of the main triangle.
However as you see in the picture there is not much change on properties of triangles.
Again Triangle APC is a isosceles triangle and PG is perpendicular to AC.
It makes Point G is a midpoint of segment AC.
Same discussion for Point F
and for Point E
So these points are still midpoints.
then Triangle EFG is still a medial triangle.
So this investigations shows that if the pedal point is on the circumcenter, Pedal triangle is also a medial triangle.