[Antreas P. Hatzipolakis]:
Let ABC be a triangle, A'B'C' the pedal triangle of H and A"B"C" the pedal triangle of I..
Denote:
(Ia), (Ib), (Ic) = the excircles of ABC, resp.
(A), (B), (C) = the excircles of A'B'C', resp
R1 = the radical axis of (Ia), (A)
R2 = the radical axis of (Ib), (B)
R3 = the radical axis of (Ic), (C)
A*B*C* = the triangle bounded by R1, R2, R3
1. ABC, A*B*C* are orthologic.
The orthologic center (ABC, A*B*C*) is the I.
The other one ?
2. IaIbIc, A*B*C* are orthologic.
The orthologic center (IaIbIc, A*B*C*) is the I.
The other one ?
3. A"B"C", A*B*C* are homothetic.
Homothetic center?
4. A"B"C", A*B*C* are orthologic.
Which is the orthologic center (A*B*C*, A"B"C") = orthocenter of A*B*C* ?
[Peter Moses]:
Hi Antreas,
A* = {2 a (a^6-a^4 b^2+a^2 b^4-b^6+2 a^4 b c+2 b^5 c-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2-4 b^3 c^3+a^2 c^4+b^2 c^4+2 b c^5-c^6),-(a-b+c) (a^6+2 a^5 b+3 a^4 b^2-a^2 b^4+2 a b^5+b^6-2 a^5 c-2 a^4 b c-4 a^3 b^2 c-2 a b^4 c-2 b^5 c-a^4 c^2-2 a^2 b^2 c^2-4 a b^3 c^2-b^4 c^2+4 a^3 c^3+4 a b^2 c^3+4 b^3 c^3-a^2 c^4+2 a b c^4-b^2 c^4-2 a c^5-2 b c^5+c^6),-(a+b-c) (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6+2 a^5 c-2 a^4 b c+2 a b^4 c-2 b^5 c+3 a^4 c^2-4 a^3 b c^2-2 a^2 b^2 c^2+4 a b^3 c^2-b^4 c^2-4 a b^2 c^3+4 b^3 c^3-a^2 c^4-2 a b c^4-b^2 c^4+2 a c^5-2 b c^5+c^6)}.
1). 2 a^7-a^6 b-2 a^5 b^2+3 a^4 b^3-a^2 b^5-b^7-a^6 c+a^4 b^2 c-a^2 b^4 c+b^6 c-2 a^5 c^2+a^4 b c^2+4 a^3 b^2 c^2+2 a^2 b^3 c^2+3 b^5 c^2+3 a^4 c^3+2 a^2 b^2 c^3-3 b^4 c^3-a^2 b c^4-3 b^3 c^4-a^2 c^5+3 b^2 c^5+b c^6-c^7::
2). a (a-b-c) (a^7 b-a^6 b^2-3 a^5 b^3+3 a^4 b^4+3 a^3 b^5-3 a^2 b^6-a b^7+b^8+a^7 c-a^5 b^2 c-2 a^4 b^3 c-a^3 b^4 c+4 a^2 b^5 c+a b^6 c-2 b^7 c-a^6 c^2-a^5 b c^2-2 a^4 b^2 c^2-2 a^3 b^3 c^2-a^2 b^4 c^2-a b^5 c^2-3 a^5 c^3-2 a^4 b c^3-2 a^3 b^2 c^3-8 a^2 b^3 c^3+a b^4 c^3+2 b^5 c^3+3 a^4 c^4-a^3 b c^4-a^2 b^2 c^4+a b^3 c^4-2 b^4 c^4+3 a^3 c^5+4 a^2 b c^5-a b^2 c^5+2 b^3 c^5-3 a^2 c^6+a b c^6-a c^7-2 b c^7+c^8)::
on lines {{10,912},{72,11507},{78,1858},{960,5248},{5123,8261},{6796,9943}}.
3). a^6-a^4 b^2+a^2 b^4-b^6+2 a^4 b c+2 b^5 c-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2-4 b^3 c^3+a^2 c^4+b^2 c^4+2 b c^5-c^6::
Denote:
(Ia), (Ib), (Ic) = the excircles of ABC, resp.
(A), (B), (C) = the excircles of A'B'C', resp
R1 = the radical axis of (Ia), (A)
R2 = the radical axis of (Ib), (B)
R3 = the radical axis of (Ic), (C)
A*B*C* = the triangle bounded by R1, R2, R3
1. ABC, A*B*C* are orthologic.
The orthologic center (ABC, A*B*C*) is the I.
The other one ?
2. IaIbIc, A*B*C* are orthologic.
The orthologic center (IaIbIc, A*B*C*) is the I.
The other one ?
3. A"B"C", A*B*C* are homothetic.
Homothetic center?
4. A"B"C", A*B*C* are orthologic.
Which is the orthologic center (A*B*C*, A"B"C") = orthocenter of A*B*C* ?
[Peter Moses]:
Hi Antreas,
A* = {2 a (a^6-a^4 b^2+a^2 b^4-b^6+2 a^4 b c+2 b^5 c-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2-4 b^3 c^3+a^2 c^4+b^2 c^4+2 b c^5-c^6),-(a-b+c) (a^6+2 a^5 b+3 a^4 b^2-a^2 b^4+2 a b^5+b^6-2 a^5 c-2 a^4 b c-4 a^3 b^2 c-2 a b^4 c-2 b^5 c-a^4 c^2-2 a^2 b^2 c^2-4 a b^3 c^2-b^4 c^2+4 a^3 c^3+4 a b^2 c^3+4 b^3 c^3-a^2 c^4+2 a b c^4-b^2 c^4-2 a c^5-2 b c^5+c^6),-(a+b-c) (a^6-2 a^5 b-a^4 b^2+4 a^3 b^3-a^2 b^4-2 a b^5+b^6+2 a^5 c-2 a^4 b c+2 a b^4 c-2 b^5 c+3 a^4 c^2-4 a^3 b c^2-2 a^2 b^2 c^2+4 a b^3 c^2-b^4 c^2-4 a b^2 c^3+4 b^3 c^3-a^2 c^4-2 a b c^4-b^2 c^4+2 a c^5-2 b c^5+c^6)}.
1). 2 a^7-a^6 b-2 a^5 b^2+3 a^4 b^3-a^2 b^5-b^7-a^6 c+a^4 b^2 c-a^2 b^4 c+b^6 c-2 a^5 c^2+a^4 b c^2+4 a^3 b^2 c^2+2 a^2 b^3 c^2+3 b^5 c^2+3 a^4 c^3+2 a^2 b^2 c^3-3 b^4 c^3-a^2 b c^4-3 b^3 c^4-a^2 c^5+3 b^2 c^5+b c^6-c^7::
2). a (a-b-c) (a^7 b-a^6 b^2-3 a^5 b^3+3 a^4 b^4+3 a^3 b^5-3 a^2 b^6-a b^7+b^8+a^7 c-a^5 b^2 c-2 a^4 b^3 c-a^3 b^4 c+4 a^2 b^5 c+a b^6 c-2 b^7 c-a^6 c^2-a^5 b c^2-2 a^4 b^2 c^2-2 a^3 b^3 c^2-a^2 b^4 c^2-a b^5 c^2-3 a^5 c^3-2 a^4 b c^3-2 a^3 b^2 c^3-8 a^2 b^3 c^3+a b^4 c^3+2 b^5 c^3+3 a^4 c^4-a^3 b c^4-a^2 b^2 c^4+a b^3 c^4-2 b^4 c^4+3 a^3 c^5+4 a^2 b c^5-a b^2 c^5+2 b^3 c^5-3 a^2 c^6+a b c^6-a c^7-2 b c^7+c^8)::
on lines {{10,912},{72,11507},{78,1858},{960,5248},{5123,8261},{6796,9943}}.
3). a^6-a^4 b^2+a^2 b^4-b^6+2 a^4 b c+2 b^5 c-a^4 c^2+2 a^2 b^2 c^2+b^4 c^2-4 b^3 c^3+a^2 c^4+b^2 c^4+2 b c^5-c^6::
on lines {{181,5905},{197,3782},{12588,14213}}.
(R (3 r^2-s^2))/(2 r (r+R)^2) X[181] + X[5905].
4). =2).
Best regards,
Peter Moses.
(R (3 r^2-s^2))/(2 r (r+R)^2) X[181] + X[5905].
4). =2).
Best regards,
Peter Moses.
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