Inverse_Pedals

Inverse pedal triangles

The pedal triangle of a point D, with respect to the triangle t = (ABC), is the triangle formed by the projections of D on the sides of t (or their prolongations, see Pedal.html ). Inverse points D, E, with respect to the circumcircle of t, define similar pedal triangles.

A proof results by comparing two sides. e.g. IK to FH. Their length ratio is the same with the diameter ratio of the two circles {KEIC} to {FDHC}. This, because the sides are seen from C and D, respectively, under the same angle. The diameters EC and CD of the corresponding circles have a ratio equal to LE/LD. This, because LC bisects angle ECD. Thus, ratio IK/HF equals EL/LD. The same is true for the other pairs of sides of the triangles (ÉJK), (HFG). This theorem has an important consequence for the 12 rotation centers (pivots) of a triangle inscribed into another. 6 of the pivots are inside the triangle and the other 6 are inverses of the previous with respect to the circumcircle of the triangle. Look at the file SixPivots.html for the corresponding picture.
A similar property holds also for the inverses with respect to the Apollonian circles. See the discussion in the file ApollonianPedalProperty.html .

Inverse pedal triangles

See Also