**The figure below shows the pedal triangle
of point P. The pedal triangle is formed from the perpendiculars to the
sides of triangle DEF from point P. The vertices of the pedal triangle,
RST, are the points of intersection of the perpendiculars with the sides
of the triangle. **

**The following is a look at the aspects
of the pedal triangle and its pedal point P with relevance to the circumcenter,
orthocenter, incenter, and centroid of triangle DEF.**

**The following is a key for the letters
given in the triangles: **

**C - circumcenter **

**H - orthocenter**

**I - incenter**

**G - centroid**

**P - pedal point**

**M1, M2, M3 - midpoints of the sides
of triangle DEF**

**R, S, T - vertices of the pedal triangle**

**D, E, F - vertices of the original triangle**

# 1 - C inside triangle | # 2 - C outside triangle | # 3 - C on side of triangle |

**In the above figures, #'s 1, 2, and
3, when the pedal point P coincides with the circumcenter C, the pedal triangle,
RST,
has become the medial triangle of DEF. Remember the circumcenter is the point equidistant
from the vertices of triangle DEF. Also the pedal triangle is formed from the perpendiculars
from P to the sides of triangle DEF. Therefore, when the pedal point P is equidistant
from the three vertices of triangle DEF, the midpoints of the sides of triangle DEF become
the points where P is perpendicular to each side, thus coinciding with the
midpoints of each side.**

# 1 - H inside triangle |
# 2 - H outside triangle |
# 3 - H on side of triangle |

**The above figures show the pedal point
P coinciding with the orthocenter of triangle DEF. **

**As noted, in figure #1, the orthocenter
is inside triangle DEF and the pedal triangle RST also remains on the interior
of DEF.**

**When the orthocenter is on the exterior
of triangle DEF, the pedal triangle also pulls to the outside of triangle DEF.**

**The orthocenter is the point of concurrency
of the altiudes of triangle DEF. Since the pedal point P coincides with the orthocenter
H, the vertices of triangle RST are in the direct line of the altitudes which
form the orthocenter. In essence, this explains why the pedal triangle remains
inside on the acute triangle (figure #1 above) and why part of the pedal
triangle rests exterior (in figure #2 above). **

**Figure #3 shows the orthocenter on a
side of triangle DEF, which is a right triangle. The orthocenter always coincides with
the vertex of the right angle in a right triangle. Here where the pedal
point P coincides with the orthocenter H, the pedal triangle becomes a straight
line. This is called the Simson line and this occurs anytime the pedal point
rests on the circumcircle of triangle DEF
which vertex F is on the circumcircle.
**

**For both the incenter (I) and the Centroid
(G), when the pedal point P coincides with those points, the pedal triangle
will always remain on the interior of triangle DEF. Since the Incenter (I)
is the point of concurrency of the angle bisectors and always remains in
the interior of triangle DEF, then the pedal triangle will also stay interior.
For the Centroid, the vertices of pedal triangle RST will remain on the sides
of triangle DEF since it was constructed to the perpendiculars of the sides of
triangle DEF. It should be clear that the pedal triangle would remain inside
for the two above since the pedal point P is inside and is constructed from
the perpendiculars to each side.**