Wednesday 3 April 2024

Extending Some Basic Geometry - Pedal Triangles and More


Somewhere in geometry class they usually introduce the concurrency theorems. Often they are introduced, then dropped almost as quickly and seldom see the light of day again. This seems somewhat a shame to me as there are some really nice generalizations of most of them. For now I want to address some extensions of perpendiculars from a point in a triangle, and some more general properties of them.
The basic class often explains that if the student takes any triangle and draws the three altitudes, they will meet in a point, often called the orthocenter of the triangle.
I cover several properties and generalizations of the orthocenter here, but one I will repeat is the neat property that if you multiply the lengths of the two sections of any altitude created by the orthocenter, the products are equal. I think of this as similar to the theorem they will (or may have already) learned about the sections of intersecting chords in a circle (actually to prove the theorem it is very helpful to know that theorem and know that the reflection of the orthocenter over any side of the triangle will always land on the circumcircle)
The Orthic triangle is a name for the triangle joining the feet of the altitudes (where they intersect the opposite side) and students may be led to discover that of all the triangles that could be inscribed in a given triangle, the one with the smallest perimeter is the orthic triangle. This has sometimes been called Fagnano's Problem since it was first posed and answered by Giovanni Francesco Fagnano dei Toschi.
A simple but pretty formula relates the lengths of the sides of the orthic triangle to the sides and angles of the original triangle. For example, in the Triangle ABC, the side of the orthic triangle nearest to vertex A is given by a*Cos (A), and likewise for the other two sides. The perimeter of the orthic triangle, then, is equal to a*Cos(A) + b*Cos(B) + c*Cos(C). I have days where I think that formula is right up there with the Pythagorean Theorem.
The Orthic triangle is also known as a pedal triangle, from the Latin word for foot, since it is the foot of the altitude. I wanted to talk today about a theorem that is very little known to students (and truth be told, to teachers of geometry) about the generalized Pedal Triangle.
In the figure above a random point P is selected in the triangle. The foot of the perpendiculars to each side from this point mark three points that are the vertices of the general pedal triangle.
Some special points and their associated pedal triangles are well known. For the orthocenter I have mentioned above some characteristics of the orthic triangle. If the incenter is chosen as the pedal point, the vertices of the pedal triangle will occur at the points where the incircle is tangent to the sides of the triangle. If the point P falls on the circumcircle of ABC, then the three feet of the perpendiculars will lie in a straight line, known as the Simson line.
There are two little known theorems about pedal triangles that are, in my mind, very beautiful. OK, actually there is only one theorem which is also generalizable to other polygons. I have to admit that I wasn't aware of the generalize version until I saw it recently on the Futility Closet blog of Greg Ross.
If you take the pedal triangle of the pedal triangle from that same point, then do that one more time (that should be ABC''' ) then the third pedal triangle will be similar to the original triangle you started with... and that happens with any random point you start with, inside or outside the original triangle.

Greg's post even had a catchy poem to describe the situation.

Begin with any point called P
(That all-too-common name for points),
Whence, on three-sided ABC
We drop, to make right-angled joints,
Three several plumb-lines, whence ’tis clear
A new triangle should appear.

A ghostly Phoenix on its nest
Brooding a chick among the ashes,
ABC bears within its breast
A younger ABC (with dashes):
A figure destined, not to burn,
But to be dropped on in its turn.

By going through these motions thrice
We fashion two triangles more,
And call them ABC (dashed twice)
And thrice bedashed, but now we score
A chick indeed! Cry gully, gully!
(One moment! I’ll explain more fully.)

The fourth triangle ABC,
Though decadently small in size,
Presents a form that perfectly
Resembles, e’en to casual eyes
Its first progenitor. They are
In strict proportion similar.

Greg credits the poem to Mary Pedoe in 1947, whom I think must be the wife of Dan Pedoe, the English-born mathematician and geometer who has authored such extraordinary books as The Gentle Art of Mathematics. I know that he married a lass named Mary Turstall.

The extension of this theorem, which was the new part to me, is that you can create a pedal rectangle or a pedal pentagon, etc by the same process and by the time you get to the n'th copy of the n-gon, "Eureka", it is similar to the original.
Simply an awesome idea, share it with your students.

I wanted to show a pedal quadrilateral progression.  They get a little messy, but notice that the nth reduction of each appears to have rotated about 180 degrees.  Which made me wonder...And after experimenting it seems that the rotation is 180 degrees.  That's from a sketch, not a geometric proof......but as the old Monkees song goes, "Now I'm a believer."

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