Pedal Triangle

By: Laura Lowe

Problem: 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 Pedal Point P.  What if P is the centroid? incenter? orthocenter? circumcenter? center of the 9 point circle?

Begin by constructing the pedal triangle for pedal point P.  (Click here to see my construction.)   Use you construction to manipulate P.  Notice that when P is outside of ABC, the pedal triangle RST is sometimes outside of ABC, as above, sometimes a degenerate triangle, and sometimes inside.

But when P is inside ABC, the pedal triangle is inside as well.

Notice also that when P is one of the vertices of ABC the pedal triangle is degenerate.

However, these are not the situations I want to look at today.  LetŐs look at what happens in the 5 cases outlined in the problem statement.

1.  The Pedal Point is the centroid of ABC.

Begin by constructing the centroid, G, of ABC.  Then construct the pedal triangle, using G as the pedal point P.  Remember that the centroid is always in the interior of ABC, so we should expect the pedal triangle to be contained with ABC whether ABC is acute or obtuse.  The acute case is above.  Here is the obtuse case.

Do you see any other relationships?  Click here to see my construction.

2.  The Pedal Point is the incenter of ABC.

It appears that P is the circumcenter of RST.

In fact, m(IR) = m(IS) = m(IT).  This is because any point on the angle bisector is equidistant from the legs of that angle.  By definition, point I is on the angle bisectors of LABC, LBCA, and LCAB.

Do you see any other relationships?  Click here to see my construction.

3.  The Pedal Point is the orthocenter of ABC.

It appears that RST is the orthic triangle of ABC.  This is because the orthic triangle is constructed from the altitudes of the triangle.  Since altitudes are perpendicular to the sides of ABC, the orthic triangle is a special case of the pedal triangle.

What if ABC is obtuse?

In this case RST is orthic to BHC for the same reason.

Do you see any other relationships?  Click here to see my construction.

4.  The Pedal Point is the circumcenter of ABC.

It looks like RST is the medial triangle of ABC.  This is because the circumcenter is constructed from the perpendicular bisectors of each side.  Therefore m(AT) = m(TC), m(AS) = m(SB), and m(BR) = m(RC).  So point R is midpoint of side BC, point T is the midpoint of side AC, and point S is the midpoint of side AB.  So RST is the medial triangle.  This does not change when ABC is obtuse.

Do you see any other relationships?  Click here to see my construction.

5.  The Pedal Point is the center of the 9 point circle of ABC.

It appears K is on the Euler Line.

In fact, it looks like K is the midpoint of the Euler Line.  Why is this?

Do you see any other relationships?  Click here to see my construction.

Some other interesting constructions:

Click here to see an animation of the locus of the midpoints of the sides of the pedal triangle when P is on the incircle.

Click here to see an animation of the locus of the midpoints of the sides of the pedal triangle when P is on the circumcircle.

Click here to see an animation of the locus of the midpoints of the sides of the pedal triangle when P is on a circle larger than the circumcircle.

Click here to see an animation of the locus of the midpoints of the sides of the pedal triangle when P is on a circle smaller than the circumcircle.