By Dorothy Evans
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
I found this investigation to
be one of the most interesting since the Pedal Point P can be anywhere on the
plane of ABC.† After creating the Pedal
Point and Pedal Triangle I chose to write up my investigation of #7 and found a
relationship between the when the
First letís look in general what happens as P moved to the side of the triangle ABC.
I first noticed that as P moved towards the sides of +ABC the point P would move towards a vertex of +RST.† Click here to try it on your own.†
While this is interesting, it also seems an obvious phenomenon since the pedal point is constructed from the perpendicular lines through point P to the sides of +ABC.
Next I decided to see what would happen if I merged the pedal point P onto a side of +ABC and animate point P along that side of the triangle.† Here you can see I have merged Point P onto segment AC.†
Notice that as P moves from the vertex A to the vertex at C the vertices R and S will slide up and down the sides of +ABC for only a part of the segment.
Here I traced the point R and S as they moved up and down while P was animated on the segment AC.† This led me to question what would happen if +ABC were obtuse and is there a relationship between the movement of R and S in relationship to +ABC.† To see P animated along segment AC click here. †
Then I made +ABC obtuse to see what the traces would look like for point S and R.† (See above)
Next I conjectured that there may be a relationship between the traces and one of the triangle centers.† My first try I created the centroid of +ABC but could not find a relationship between the traces and the centroid.† Next, I tried the orthocenter since it is constructed from the perpendiculars through each vertex to the opposing side of the triangle.† Here I found a relationship between the orthocenter and the Pedal Point P when itís on a side of the triangle.†
Here is the acute +ABC.†††††††††††††††††††††††††††††††††††††† Here is the obtuse +ABC.†
Notice for both of these (the acute and obtuse) as the pedal point P moves along the side of the triangle.† The other two vertices of the pedal triangle (R and S) move along the segment between the vertices of +ABC and where the altitude intersects on the other side of +ABC.† To see the animation click here.†
Furthermore in answer to question 8.† What if P is one of the vertices of triangle ABC?† In this case the pedal triangle collapses into a line.†