Denote:
Ab, Ac = The circumcenters of APB', APC', resp.
Bc, Ba = The circumcenters of BPC', BPA', resp.
Ca, Cb = The circumcenters of CPA', CPB', resp.
1. M1a,M1b,M1c = The midpoints of AbAc,BcBa,CaCb, resp.
Which is the locus of P such that:
1.1. ABC, M1aM1bM1c are perspective?
1.2. ABC, M1aM1bM1c are orthologic?
1.3. The perpendicular bisectors of AbAc,BcBa,CaCb are concurrent?
For P = G:
1.2. ABC, M1aM1bM1c are orthologic.
Orthologic center (M1aM1bM1c, ABC) = N
Orthologic center (ABC, M1aM1bM1c) : Anopolis #1284, #1295
1.3. The perpendicular bisectors concur at van Lamoen Circle Center X(1153)
2. M2a,M2b,M2c = The midpoints of BcCb, CaAc, AbBa, resp.
Which is the locus of P such that:
2.1. ABC, M2aM2bM2c are perspective?
2.2. ABC, M2aM2bM2c are orthologic?
2.3. The perpendicular bisectors of BcCb, CaAc, AbBa are concurrent?
For P = G:
2.2. ABC, M2aM2bM2c are orthologic.
Orthologic center (M2aM2bM2c, ABC) = ?
Orthologic center (ABC, M2aM2bM2c) = G
2.3. The perpendicular bisectors concur at van Lamoen Circle Center X(1153)
3. M3a,M3b,M3c = The midpoints of BaCa, CbAb, AcBc, resp.
Which is the locus of P such that:
3.1. ABC, M3aM3bM3c are perspective?
3.2. ABC, M3aM3bM3c are orthologic?
3.3. The perpendicular bisectors of BaCa, CbAb, AcBc are concurrent?
For P = G
3.2. ABC, M3aM3bM3c are orthologic.
Orthologic center (M3aM3bM3c, ABC) = O
Orthologic center (ABC, M3aM3bM3c) = ?
3.3. The perpendicular bisectors concur at van Lamoen Circle Center X(1153)
4. Which is the locus of P such that:
4.1. M1aM1bM1c, M2aM2bM2c
4.2. M1aM1bM1c, M3aM3bM3c
4.3. M2aM2bM2c, M3aM3bM3c
are perspective/orthologic ?
4.4. The Euler lines of M1aM1bM1c, M2aM2bM2c, M3aM3bM3c are concurrent?
For P = G ??
5. Which is the locus of P such that:
4.1. M1aM2aM3a, M1bM2bM3b
4.2. M1aM2aM3a, M1cM2cM3c
4.3. M1bM2bM3b, M1cM2cM3c
are perspective/orthologic ?
4.4. The Euler lines of M1aM2aM3a, M1bM2bM3b, M1cM2cM3c are concurrent?
For P = G ??
Antreas P. Hatzipolakis, 4 May 2014
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