Assignment #9
Podal Triangles Explorations
by
Victor
L. BrunaudVega






Podal triangle when
point P belongs to one of the sides of ΔABC 




Constructing the podal triangle of ΔABC


In any triangle ABC, as in these
picture, given some point P (which can be interior or exterior to the
triangle), if we draw the three perpendicular lines from that point P to the
three sides of the triangle (or to the projections of the sides, if it is the
case), determining the points P1, P2 and P3. 






Tracing segments among these points we
have a new triangle, the PODAL triangle of ΔABC 












Podal triangle when point P belongs to one of the sides of ΔABC 



When the point P belongs to the side AB
of the triangle ABC, like in the picture of the left, both point P and point
P1 are coincident. If the point
P belongs to the side BC of the triangle ABC, both point P and point P2 are
coincident. And if the point P
belongs to the side CA of the triangle ABC, then both point P and point P3
are coincident. But the interesting effect of these
situations is the relation between the angles. You can see relationships among some measurement of angles
here. 





Another proof of this relationship is
developed now. 




















Podal triangle when point P belongs to ΔABCÕs circumcircle 






This is beautiful. If the point P belongs to the
circumcircle of the triangle ABC, the points P1, P2 and P3 become part of the
same line and the podal triangle disappears. You can see an animation of this case here. As the point P moves following the
circumcircle of the triangle ABC counterclockwise, the segment to which
belongs P1, P2 and P3 spins clockwise, and vice versa. 




If we trace the segment to which the
points P1, P2 and P3 belong during the movement of point P on the
circumcircle of the triangle ABC, we will get a rare shape, called tricuspid deltoid, relative to the deltoid
muscle in the human shoulder. 
































Podal triangle when point P belongs to the ΔABCÕs incircle 






I was exploring possibilities moving the
point P along the incircle of the triangle ABC, but I could not find much new
stuff. You can play an animation
here. Even thought, the point P stays into the
interior region of the triangle ABC and whenever the point P reaches one of
the sides, it coincides with the correspondent point P1, P2 or P3. 




























