Πέμπτη, 22 Δεκεμβρίου 2011

Reflections of cevians

Let ABC be a triangle and A'B'C' the cevian triangle of H.


Denote:

12 := the reflection of AA' in BB'
13 := the reflection of AA' in CC'

23 := the reflection of BB' in CC'
21 := the reflection of BB' in AA'

31 := the reflection of CC' in AA'
32 := the reflection of CC' in BB'

1'2' := the parallel to 12 through B'
1'3' := the parallel to 13 through C'

2'3' := the parallel to 23 through C'
2'1' := the parallel to 21 through A'

3'1' := the parallel to 31 through A'
3'2' := the parallel to 32 through B'

A* := 1'2' /\ 1'3'
B* := 2'3' /\ 2'1'
C* := 3'1' /\ 3'2'

1. The triangles ABC, A*B*C* are perspective (at H)
2. The triangles A'B'C', A*B*C* are perspective (at H)
2. The points A'B'C'A*B*C* are concyclic (on the NPC)

Generalization:

Point P instead of H.


Which is the locus of P such that:

1. The triangles ABC, A*B*C* are perspective ?
2. The triangles A'B'C', A*B*C* are perspective ?
2. The points A'B'C'A*B*C* are conconic ? (when the conic is circle) ?

Variation:

1'2' := the parallel to 12 through C'
1'3' := the parallel to 13 through B'

2'3' := the parallel to 23 through A'
2'1' := the parallel to 21 through C'

3'1' := the parallel to 31 through B'
3'2' := the parallel to 32 through A'


APH 22 December 2011

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