The pedal triangle of a point E, with respect to the triangle t = (ABC), is the triangle formed by the projections of E1, E2, E3 on the sides of t (or their prolongations see Pedal.html ). Consider the Apollonian circle c(D,|DA|), defined as the locus of points P such that |PA|/|PB| = |AB|/|AC|.
[1] Let F be the inverse of E with respect to (c) and E1E2E3 , F1F2F3 the corresponding pedals. Then the length-ratio of homologous sides (f.e. |E2E3|/|F2F3|) is equal to the ratio |AE|/|AF| = |A1E|/|A1F| for all such pairs of sides.
[2] Triangles E1E2E3 and F1F2F3 are similar and inversely oriented.
[1] That |E2E3|/|F2F3| = |AE|/|AF| is easily seen from the cyclic quadrilaterals {AE2EE3} and {AF3FF2}. E2E3 and respectively F2F3 are chords of the circumcircles of these quadrilaterals seen from A under the same angle, thus their lengths being in ratio as the diameters of these circles, which are precisely |AE| and |AF| correspondingly.
The quotient of the other sides needs some more work. For example |E1E2|/|F1F2| = (|BE|*sin(B))/(|CF|*sin(C)) = (*), and one needs the relation of the lengths of segments whose endpoints are inverse with respect to (c) (see InverseLengths.html ), as is the case with CF and BE.
In fact, by the sine theorem for triangles, (*) = (|BE|*b)/(|CF|*c). By the aforementioned reference:
|BE|/|CF| = (r2/(|DF|*|DC|)), b/c = r/|DB| => (*) = r3/(|DF|*|DC|*|DB|)= r/|DF| = |AE|/|AF|.
[2] is an immediate consequence of [1].
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Apollonian pedal property ![[0_0]](ApollonianPedalProperty_files/ApollonianPedalProperty_0_0_0.png)
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