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

Pedal Triangles and Iteration

Given a point $P$ inside triangle $ABC$, the pedal triangle of $P$ is the triangle formed by projecting $P$ perpendicularly onto each side of $ABC$.

Construction

• $D_1$ = foot of perpendicular from $P$ to $BC$
• $E_1$ = foot of perpendicular from $P$ to $CA$
• $F_1$ = foot of perpendicular from $P$ to $AB$

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.

Triangle and Interior Point

Start with triangle $ABC$ 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 $BC$
• $E_1$ is the foot on $CA$
• $F_1$ is the foot on $AB$

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 $ABC$.

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 $\cos \alpha \cdot \cos \beta \cdot \cos \gamma < 1$, guaranteeing geometric convergence.

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

Conclusion

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 $\cos \alpha \cdot \cos \beta \cdot \cos \gamma$ where $\alpha$, $\beta$, $\gamma$ are angles subtended at $P$.