Pedal Triangles and Iteration

An exploration of pedal triangles: projecting a point onto each side of a triangle to form a pedal triangle, then iterating the process to produce smaller nested triangles converging toward the pedal point.

Keywords: pedal triangle, projection, perpendicular foot, iteration, convergence

Difficulty: advanced

Given a point $P$ inside triangle $A B C$ , its pedal triangle is formed by projecting $P$ perpendicularly onto each side.

Strategy: drop the three perpendiculars from $P$ to the sides to get the feet $D_{1}$ , $E_{1}$ , $F_{1}$ , then iterate the construction on each new pedal triangle. The animation shows the nested triangles shrinking toward $P$ .

Triangle and Interior Point

Start with triangle $A B C$ and an interior point $P$ . The pedal triangle is formed by projecting $P$ perpendicularly onto each side.

Perpendicular Feet $D_{1}$ , $E_{1}$ , $F_{1}$

Project $P$ perpendicularly onto each side of the triangle:

$D_{1}$ is the foot on $B C$
$E_{1}$ is the foot on $C A$
$F_{1}$ is the foot on $A B$

Each projection creates a right angle at the foot.

The First Pedal Triangle

Connect the three feet $D_{1}$ , $E_{1}$ , $F_{1}$ to form the first pedal triangle.

This triangle is always contained within the original triangle $A B C$ .

The Second Pedal Triangle

Now project $P$ onto the sides of the first pedal triangle $D_{1} E_{1} F_{1}$ :

$D_{2}$ is the foot on $D_{1} E_{1}$
$E_{2}$ is the foot on $E_{1} F_{1}$
$F_{2}$ is the foot on $F_{1} D_{1}$

Connect them to form the second pedal triangle $D_{2} E_{2} F_{2}$ , which is even smaller.

Convergence Toward $P$

Each iteration produces a smaller pedal triangle nested inside the previous one. The sequence of triangles converges toward the point $P$ .

Why it shrinks: Each perpendicular projection reduces the distance from $P$ to the triangle. The ratio of successive pedal triangles involves $cos α \cdot cos β \cdot cos γ < 1$ , guaranteeing geometric convergence.

Continuing the iteration infinitely, the triangles shrink to the single point $P$ .

Iterating the pedal triangle construction produces a sequence of nested triangles that shrink toward $P$ .

Each pedal triangle is similar to one related to the original, and the shrinkage ratio depends on $cos α \cdot cos β \cdot cos γ$ where $α$ , $β$ , $γ$ are angles subtended at $P$ .

Notes

Construction

$D_{1}$ = foot of perpendicular from $P$ to $B C$
$E_{1}$ = foot of perpendicular from $P$ to $C A$
$F_{1}$ = foot of perpendicular from $P$ to $A B$

The triangle $D_{1} E_{1} F_{1}$ is the first pedal triangle.

Iteration

We can repeat the process: project $P$ onto the sides of the pedal triangle $D_{1} E_{1} F_{1}$ to get a second pedal triangle $D_{2} E_{2} F_{2}$ .

Each successive pedal triangle is smaller than the previous one, and the sequence of triangles converges toward $P$ .

Why It Shrinks

Each perpendicular projection reduces distances. The pedal triangle is always contained within the previous triangle, and its vertices are strictly closer to $P$ . The ratio of successive triangles is related to $cos^{2}$ of angles, ensuring geometric convergence.