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
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