21994TCS: P6 - P10
- Jul 25, 2013[Hatzipolakis - F. Javier]
Let ABC be a triangle, P = (x:y:z) a point and A'B'C'
the pedal triangle of P.
A* := (The Reflection of BC in BP) /\ (The Reflection of BC in CP)
B* := (The Reflection of CA in CP) /\ (The Reflection of CA in AP)
C* := (The Reflection of AB in AP) /\ (The Reflection of AB in BP)
[The triangles A*BC, B*CA, C*AB share the same incenter P]
Oa := The Circumcenter of A*BC
Ob := The Circumcenter of B*CA
Oc := The Circumcenter of C*AB
P* := The Point of Concurrence of the Circumcircles
of A*BC, B*CA, C*AB
[We have seen recently this point in Hyacinthos]
La := The Reflection of P*Oa in B'C'
Lb := The Reflection of P*Ob in C'A'
Lc := The Reflection of P*Oc in A'B'.
The Triangles ABC, Triangle bounded by (La,Lb,Lc)
are parallelogic for P on the Euler line.
One of the parallelocic centers lies on the circumcircle.
Coordinates of Parallelogic centers on the circumcircle
for P = H, O, G, N and midpointGH in a File in the Reference: