HYACINTHOS 26595

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle.

The perpendicular to AB at A intersects BC at A1
The perpendicular to BC at B intersects CA at B1
The perpendicular to CA at C intersects AB at C1

Let L1, M1, N1 be the midpoints of AA1, BB1, CC1 resp.

Let  A* =  BL1 /\ CM1, Β* = CM1 /\ AN1, C* = AN1 /\ BL1.

Then the lines A1A*, B1B*, C1C* are concurrent at a point V1

 

Source: Kostas Vittas, Romantics of Geometry 1165

 

Similarly:

Let ABC be a triangle.

The perpendicular to AC at A intersects BC at A2
The perpendicular to BA at B intersects CA at B2
The perpendicular to CB at C intersects AB at C2

Let L2, M2, N2 be the midpoints of AA2, BB2, CC2 resp.

Let  A** =  BN2 /\ CL2 , Β** = CL2 /\ AM2, C** = AM2 /\ BN2  

Then the lines A2A**, B2B**, C2C** are concurrent at a point V2

V1, V2 are bicentric points (?).
Which point is the midpoint V of V1V2  ?



[Peter Moses]:

 

Hi Antreas,

>V1, V2 are bicentric points (?).

Yes.
V1 = {a^2 (a^2-b^2+c^2) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2+6 a^2 b^2 c^2+5 b^4 c^2+3 a^2 c^4-3 b^2 c^4-c^6),b^2 (a^2+b^2-c^2) (-a^6+3 a^4 b^2-3 a^2 b^4+b^6-3 a^4 c^2+6 a^2 b^2 c^2-3 b^4 c^2+5 a^2 c^4+3 b^2 c^4-c^6),c^2 (-a^2+b^2+c^2) (-a^6+5 a^4 b^2-3 a^2 b^4-b^6+3 a^4 c^2+6 a^2 b^2 c^2+3 b^4 c^2-3 a^2 c^4-3 b^2 c^4+c^6)}.

>Which point is the midpoint V of V1V2  ?

a^2 (a^14-4 a^12 b^2+3 a^10 b^4+8 a^8 b^6-17 a^6 b^8+12 a^4 b^10-3 a^2 b^12-4 a^12 c^2+12 a^10 b^2 c^2-2 a^8 b^4 c^2-16 a^6 b^6 c^2+8 a^4 b^8 c^2+4 a^2 b^10 c^2-2 b^12 c^2+3 a^10 c^4-2 a^8 b^2 c^4-6 a^6 b^4 c^4-20 a^4 b^6 c^4+19 a^2 b^8 c^4+6 b^10 c^4+8 a^8 c^6-16 a^6 b^2 c^6-20 a^4 b^4 c^6-40 a^2 b^6 c^6-4 b^8 c^6-17 a^6 c^8+8 a^4 b^2 c^8+19 a^2 b^4 c^8-4 b^6 c^8+12 a^4 c^10+4 a^2 b^2 c^10+6 b^4 c^10-3 a^2 c^12-2 b^2 c^12):: 

on lines {{647,11424},{1971,9605},...}.

Best regards,
Peter Moses.