Where to Put Your Pedal Point
An Exploration of Pedal Triangles
The following investigation is based on prompts from Exploration 9, all of which pertain to the pedal triangle. Click here for the script tool associated with this construction.
Given a triangle, ABC, and a point, P, in the same plane, we can construct the pedal triangle by constructing perpendicular lines from the pedal point, which is P, to the sides of the triangle. The intersections result in the vertices of the pedal triangle:
We have three different scenarios for the location of P. The first is when
P is located inside of triangle ABC:
This pedal triangle will always lie inside of ABC. This is because the vertices of the pedal triangle are dependent on the perpendiculars to the sides of ABC. Since P is inside the triangle, the perpendiculars will always intersect at the actual sides of the triangle and not an extension of these segments. Therefore, the pedal triangle will always lie inside ABC:
The second scenario is when P lies outside of triangle ABC:
This is a little trickier to generalize, but there are some notable characteristics associated with the perpendiculars. Let’s say that P lies to the left of segment AB, like in the figure above. We know that one of the vertices of the pedal triangle will normally lie on AB. This is because for every point lying on a segment, there is one and only one perpendicular that passes through this segment’s point.
The other vertices can lie inside or outside ABC depending on the location of P:
However, this is not the case when P lies in a location such that AB must be extended to construct the perpendicular. In this case, the whole pedal triangle will either lie completely outside of ABC or intersect ABC:
This is because at least one of the sides of triangle ABC must be extended past AB to construct the pedal triangle:
It is possible in this situation for the three intersection points to form a segment.
This, of course, occurs when all three intersection points lie on the same line. Obviously, there is no way to draw a triangle through three points lying on the same line because the angle measure is already 180.
The third scenario is when P lies on one of the sides of ABC:
This scenario is also tricky to generalize. The only characteristic that we know for sure is that P will be one of the vertices of the pedal triangle because only one perpendicular can be constructed on any point lying on a segment. The other vertices of the pedal triangle are dependent on the angles of ABC, in regards to whether or not they lie inside or outside ABC:
As the above images show, the trend seems to be that when ABC is an obtuse triangle, one of the vertices of the pedal triangle will lie outside of ABC if the pedal point P lies on one of the sides of ABC.
What if P is one of the vertices of ABC? This will always result in a segment instead of a triangle.
Why is this? Like previously stated, a perpendicular can be drawn through any point on a segment. However, a vertex, in technicality, lies on two sides of the triangle. Therefore, two of the perpendiculars can be drawn through this point. Once we find the other intersection point, we only have two points of intersection, so only a segment can be drawn between these two points.
What if pedal point P is the incenter of triangle ABC? Then the vertices of the pedal triangle will lie on the three points that are concurrent between triangle ABC and the incircle of the triangle.
Why is this? The incenter is constructed by finding the three angle bisectors of ABC and noting their intersection point. This intersection is also equidistant from all three sides of ABC because all points equidistant from the center of a circle form the circle. We can find these distances to the sides of ABC by finding segments extending from the incenter that are perpendicular to the sides of the triangle. If we equate the incenter to point P, when we find these perpendicular distances, we are forming the vertices of the pedal triangle.
What if P is the orthocenter? Then the pedal triangle is actually the orthic triangle of ABC.
Why is this? The orthocenter is the point of intersection of the altitudes of triangle ABC. Altitudes are segments that are perpendicular to each side. Therefore, when we equate the orthocenter to point P, we draw the altitudes, which all pass through P and are perpendicular to the sides of ABC. This is essentially constructing the pedal triangle.
What if the orthocenter is outside of triangle ABC? It’s still the same as finding a pedal triangle with our point P outside of ABC. We simply just extend the sides of the triangle and find the altitudes that way. It will still result in the orthic triangle.
What if P is the circumcenter of triangle ABC? The resulting pedal triangle is the medial triangle of ABC.
Why is this? When we construct the circumcenter, we first find the midpoints of each side of ABC and then construct perpendicular lines through these midpoints. Therefore, the line that intersects both the circumcenter and the side of the triangle must be perpendicular to the side of the triangle in which it intersects. This is the same process as finding any ordinary pedal triangle except the perpendiculars will always intersect the midpoints. When we connect all three of these midpoints, we create the medial triangle
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