CEVIANS, RADICAL CENTERS, EULER LINE

Let ABC be a triangle, P a point, A'B'C' the cevian triangle of P and Ma, Mb, Mc points on AA',BB',CC', resp.

Denote:

X = the radical center of the ciecles (Ma, MaB), (Mb, MbC), (Mc, McA)

Y = the radical center of the circles (Ma, MaC), (Mb, MbA), (Mc, McB)

M = the midpoint of the line segmant XY

1.

Let Ma, Mb, Mc be points such that: MaA / MaA' = MbB / MbB' = McC / McC' = t

Which is the locus of M as t varies?

For P = G, the locus is the Euler line.

For t = -1 (ie Ma, Mb, Mc = the midpoints of AA', BB', CC', resp.)

==> M is the circumcenter O of ABC for all P's.

2. Let Ma, Mb, Mc be points such that: MaA / MaP = MbB / MbP = McC / McP = t

Which is the locus of M as t varies?

For P = O ==> M = the NPC center N

For P = G ==> The locus is the Euler Line.

3. Let Ma, Mb, Mc be points such that: MaP / MaA' = MbP / MbB' = McP / McC' = t

Which is the locus of M as t varies?

For P = G ==> the locus is the Euler line.

Antreas P. Hatzipolakis, 31 March 2013