Τρίτη 22 Οκτωβρίου 2019

HYACINTHOS 24449

 
Let ABC be a triangle, NaNbNc the pedal triangle of N and OaObOc the pedal triangle of O.

Denote:

N1, N2, N3 = the reflections of N in BC, CA, AB, resp.
O1,O2, O3 = the reflections of O in BC, CA, AB, resp.
 
1. O1O2O3, N1N2N3 are perspective. The perspector lies on the circumcircle.

2. The parallels to O1N1, O2N2, O3N3 through Oa,Ob,Oc, resp. are concurrent.

3. The parallels to O1N1, O2N2, O3N3 through Na,Nb,Nc, resp. are concurrent.

 

[Peter Moses]:

Hi Antreas,
 
>1. O1O2O3, N1N2N3 are perspective. The perspector lies on the circumcircle.
X(110).
 
>2. The parallels to O1N1, O2N2, O3N3 through Oa,Ob,Oc, resp. are concurrent.
X(1511).
 
>3. The parallels to O1N1, O2N2, O3N3 through Na,Nb,Nc, resp. are concurrent.
(2 a^4-a^2 b^2-b^4-a^2 c^2+2 b^2 c^2-c^4) (a^6-3 a^4 b^2+3 a^2 b^4-b^6-3 a^4 c^2-a^2 b^2 c^2+b^4 c^2+3 a^2 c^4+b^2 c^4-c^6)::
 
On lines {{2,399},{3,146},{4,7666},{5,4 9},{30,113},{69,10201},{74,549 },{125,3628},{140,5663},{403,3 043},{468,1986},{495,10091},{4 96,10088},{542,547},{546,9820} ,{548,2777},{550,7728},{597, 9976},{1125,2771},{1154,10096} ,{1656,3448},{2931,5654},{ 2948,5886},{3564,6593},{3582, 6126},{3584,7343},{3850,10113} ,{5055,9143},{5066,7687},{ 5432,7727},{5876,10125},{5898, 7693},{6140,6592},{6153,10095} ,{6677,9826},{7525,10117},{754 2,7723},{7722,10018}}.
Complement of the complement of X(399).
 
Midpoint of X(i) and X(j) for these {i,j}: {{5,110},{113,1511},{125,5609} ,{549,5655},{550,7728},{6053,6 699}}.
Reflection of X(i) in X(j) for these {i,j}: {{125,3628},{140,5972},{10113, 3850}}.
Crossdifference of every pair of points on line {2081,2433}.
{X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): {(113,5642,1511), (5972,6053,6699)}.
3 X[3] + X[146], 3 X[5] - X[265], 3 X[110] + X[265], 3 X[2] + X[399], X[74] - 3 X[549], 3 X[113] - X[1539], 3 X[1511] + X[1539], 5 X[1656] - X[3448], 2 X[3628] + X[5609], X[1511] - 3 X[5642], X[113] + 3 X[5642], X[1539] + 9 X[5642], X[2931] + 3 X[5654], X[74] + 3 X[5655], X[2948] + 3 X[5886], 3 X[5972] + X[6053], 3 X[140] + 2 X[6053], 3 X[140] - 2 X[6699], 3 X[5972] - X[6699], 3 X[5066] - 2 X[7687], 3 X[5055] + X[9143], 3 X[597] - X[9976].
 
Best regards,
Peter Moses.

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