*by Soo Jin
Lee*

1a. Let triangle ABC be any triangle. Then
if P is any point in the plane, then the triangle formed by constructing
perpendiculars to the sides of ABC (extended if necessary) locate
three points R, S, and T that are the intersections. Triangle
RST is the ** Pedal Triangle **for

For any given triangle ABC, take arbitrary point p(pedal point)
at the outside of the triangle.

then construct perpendicular lines from the pedal point to the each of the lines.

Now we have triangle RST which is the PEDAL TRIANGLE for PEDAL POINT P.

The interesting thing here is that whenever we move pedal point P pedal triangle RST also move together.

let's investigate our exploration to the different circumstances of pedal point P

2. let's suppose that our point P is the centroid of our triangle ABC. what happens to our Pedal Triangle?

If P is the centroid, then our pedal triangle is created on the interior of ABC and remains the same triangle because since P is the centroid, it remains the same unless our triangle is changed. So in conclusion, our pedal triangle will never move outside of our triangle ABC.

3.Let's develop our investigation to find pedal triangle when pedal point point p is incenter of ABC.

what we have here is another triangle inside of our original ABC. Also once again, our triangle remains in the same place and never moves because our point P is in the incenter and will also never move. So no matter what, a pedal triangle, when it's pedal point is the incenter of triangle ABC, will be a stationary triangle on the inside of our triangle ABC.

4. Let's investigate what happens when our pedal point P is also the orthocenter of the triangle.

we see that our pedal pint p is the orthocenter of our triangle ABC. our pedal triangle RST is once again located inside of triangle ABC. From investigations on GSP, we know that because the pedal point is also the orthocenter, it will never change unless our original triangle ABC changes. so this triangle, pedal triangle, will never move outside of ABC.

5. what if P is the circumcenter?

Once again, we see that our pedal triangle RST is inside of our triangle ABC. Also again, we see that no matter what, because our pedal point P is also the circumcenter, our pedal triangle will never move unless P moves or is moved.

6. What if . . . P is the Center of the nine point circle for triangle ABC?

As we can see, once again our pedal triangle is inside of ABC, and it never goes outside of the triangle ABC. However what I have find interesting is that whenever I move my points A, B or C close to the nine point center, the pedal triangle gradually disappears.

7. What if P is on a side of the triangle?

Once again our pedal triangle is inside of ABC.
It is possible, however, for this triangle to move outside of
ABC. As the point P or rather as point C of ABC is extended out,
our pedal triangle begins to also move outward. See below how
the pedal triangle is beginning to move outside of ABC while point
P is still located on the side in the same place.

8. What if P is one of the vertices of triangle
ABC?

As we can see above, we no longer get the triangle instead we get the segment.

9. Find all conditions in which the three vertices
of the Pedal triangle are colinear (that is, it is a degenerate
triangle). This line segment is called the ** Simson Line.**Now we want to investigate something called the simson
line. This is where all of the three vertices of the pedal triangle
are collinear. From our investigation on GSP, we see that as P
approaches one of the vertices or becomes one of the vertices
(as in the above example) of triangle ABC we do indeed get a line.
This line can be called the simson line. The pictures below show
what happens as P is approaching the vertices.

as P goes to the A

as P goes to the B,

as P goes to the C,

10. Locate the midpoints of the sides of the Pedal Triangle. Construct a circle with center at the circumcenter of triangle ABC such that the radius is larger than the radius of the circumcircle. Trace the locus of the midpoints of the sides of the Pedal Triangle as the Pedal Point P is animated around the circle you have constructed. What are the three paths?

the trace will turn out like the picture below;

If you want to see the proccess of getting these paths, please click here.