The orthic triangle and the medial trianglee are two instances of a more general type of associated triangle. Let P be any point inside a given triangle ABC, and let perpendiculars PA1,PB2.PC3 be dropped to the three sides BC, CA, AB, as below figure. The feet of these perpendiculars are the vertices of a triangle A1B1C1 which is called the pedal triangle of triangle ABC for the pedal point P. The restrction of P to interior positions can be relaxed if we agree to insist that P shall not lie on the circumcircle of triangle ABC. Clearly, the orthic triangle or the medial triangle arises when P is the orthocenter or the circumcenter, respectively.
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Let us examine the above figure more closely.
the right angles at B1 and C1 indicate that these points lie on
the circle with diameter AP; in other words, P lies on the circumcircle
of triangle AB1C1. Applying the law of sines to this triangle
and also to triangle ABC itself, we obtain
whence
Similary,
We proved thus proved;
Theorem:
If the pedal point is distant x,y,z from the vertices of triangle
ABC, the pedal triangle has sides
The case when x=y=z=R is of course, familiar.
An interesting exercise involving pedal triangles is at the same time a delightful example of imagination in geometry. In the following figure, an interior point P has been used to determine triangle A1B1C1, the (first) pedal triangle of triangle ABC. The same pedal point P has been used again to determine triangle A2B2C2, the pedal triangle of triangle A1B1C1,which we naturally call the "second pedal triangle" of triangle ABC. A third operation yields triangle A3B3C3, the pedal triangle of triangle A2B2C2. The understanding is that, for this " third pedal triangle" also, we use the same pedal point P.
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In this terminology, Neuberg's discovery can be expressed thus:
Theorem: The third triangle is similar to the
original triangle.
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