Given a triangle ABC and a point P the circumcevian triangle of P with respect to ABC is triangle DEF formed by the intercepts D, E, F of the cevians AP, BP, CP of P with the circumcircle.
The main property of the circumcevian is that it is similar to the corresponding pedal triangle of P with respect to ABC. Below the pedal of P is GHI. The equality of angles indicates the proof of the property.
The question of finding all triangles A'B'C' circumcevian with ABC (in other words perspective to ABC and inscribed in the circumcircle of ABC) and similar to a fixed triangle A0B0C0 can be reduced to the question of finding all pivots of A'B'C' inside ABC. There are in general 12 solutions. This is handled in the file SixPivots.html .
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Circumcevian triangle ![[0_0]](Circumcevian_files/Circumcevian_0_0_0.png)
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![[1_0]](Circumcevian_files/Circumcevian_0_1_0.png)
![[1_1]](Circumcevian_files/Circumcevian_0_1_1.png)