Let ABC be a triangle and
1. A'B'C' the pedal triangle of H (orthic triangle)
Denote:
(Oa) = the circumcircle of OBC.
(O1) = the reflection of (Oa) in BC.
(O'1) = the reflection of (O1) in HA'.
(Ob) = the circumcircle of OCA.
(O2) = the reflection of (Ob) in CA.
(O'2) = the reflection of (O2) in HB'.
(Oc) = the circumcircle of OAB.
(O3) = the reflection of (Oc) in AB.
(O'3) = the reflection of (O3) in HC'.
The circumcenter of the triangle O'1O'2O'3 lies on the OH line (Euler line)
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2. A'B'C' the pedal triangle of I.
Denote:
(Oa) = the circumcircle of IBC.
(O1) = the reflection of (Oa) in BC.
(O'1) = the reflection of (O1) in IA'.
(Ob) = the circumcircle of ICA.
(O2) = the reflection of (Ob) in CA.
(O'2) = the reflection of (O2) in IB'.
(Oc) = the circumcircle of IAB.
(O3) = the reflection of (Oc) in AB.
(O'3) = the reflection of (O3) in IC'.
The circumcenter of the triangle O'1O'2O'3 lies on the OI line
Generalizations (Loci):
Let ABC be a triangle, P,P* two isogonal conjugate points and A'B'C',A"B"C" the pedal triangles of P,P*.
Denote:
(Oa) = the circumcircle of PBC.
(O1) = the reflection of (Oa) in BC.
(O'1) = the reflection of (O1) in PA'.
(O"1) = the reflection of (O1) in P*A"
(Ob) = the circumcircle of PCA.
(O2) = the reflection of (Ob) in CA.
(O'2) = the reflection of (O2) in PB'.
(O"2) = the reflection of (O2) in P*B"
(Oc) = the circumcircle of PAB.
(O3) = the reflection of (Oc) in AB.
(O'3) = the reflection of (O3) in PC'.
(O"3) = the reflection of (O3) in P*C"
R' = the circumcenter of O'O'2O'3
R" = the circumcenter of O"1O"2O"3
Which is the locus of P such that:
1. O, P, R'
2. O, P, R"
3. O, R', R"
4. P, P*, R'
5. P, R', R"
are collinear ?
The McCay cubic?
Antreas P. Hatzipolakis, 16 April 2014.
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