# The Final Pathetic Bleatings of the Forum

Question:
I humbly request the wisdom of O Sensei

O Sensi, what is the meaning of life?

Replies:

Life is like a theorem about triangles. Allow me to show you a few.

The bisectrix of an angle and the bisectrix of the opposite side of a triangle intersect on the circumcircle.

The inscribed circle touches the circle passing through the midpoints of the sides.

The leg of an altitude is the midpoint of the segment between the orthocenter and the intersection of the altitude with the circum-circle.

The triangle formed by the trisectrices of adjacent angles of an arbitrary triangle is equilateral.

The three segments connecting each vertex with the point of tangency between the opposite side and the inscribed circle are concurrent. The point at which these segments intersect is called the Gergonne point.

A triangle is congruent to the triangle whose vertices are the reflections of the circum-center accross the sides of the triangle.

The two segments on a side of a triangle detrmined by the points of tangency of the incircle and the excircle are congruent.

The centers of the equilateral triangles erected on the sides of a triangle form an equilateral triangle, called the Napoleon triangle.

Huh, huh, huh, he said "erected".

Amazingly, the centers of equilateral triangles erected on the other sides of the sides of a triangle form an equilateral triangle again.

Huh, huh, he said "erected" again.

The three segments connecting each vertex with the point of tangency between the opposite side and the opposite exscribed circle are concurrent. The point of intersection is called Nagel's point.

That picture does not show the details.

I would do everything for you, Gwyneth.

Let me tell you something about pedal triangles. Pick a point inside a triangle and from it drop perpendiculars to the sides of the triangle. The legs of the perpendiculars form a triangle. This triangle is called the pedal triangle (for a given point).

So? Where is the theorem?

You must be patient. Now, we can take the pedal triangle and the same point, and draw the pedal triangle of the pedal triangle.

Hmmmm. Ok. But what's the point?

Next, we draw the third pedal triangle, which is the pedal triangle of the pedal triangle of the pedal triangle.

Would you please stop masturbating with your "pedal" triangles and gives us a theorem?

Yes, of course. The third pedal triangle is similar to the original triangle.

Wohooo. You are my hero, Andrej.

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