ADGEOM 4604 * ADGEOM 4608

#4604

 

Dear geometers,

 

Let ABC be a triangle with circumcenter O.

 

P is on its Euler line.

 

Oa, Ob, Oc are isogonal conjugate of O wrt triangles PBC, PCA, PAB.

 

Then centroid of OaObOc lies on Euler line of ABC.

 

Which is this point in term of P?

 

Best regards,

Tran Quang Hung.

 

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#4608

 

Dear Tran Quang Hung.

If P is on its Euler line such that OP:PH=t , then centroid Q of
OaObOc ( lies on Euler line of ABC) is:

Q = (a^2-b^2) (a^2-c^2) (a^4-(b^2-c^2)^2) t (a^16 t^2-(b^2-c^2)^6
(b^2+c^2)^2 t^2-a^14 (b^2+c^2) t (1+3 t)+a^2 (b^2-c^2)^4 (b^2+c^2) t (-2
b^2 c^2 (-2+t)+b^4 (1+3 t)+c^4 (1+3 t))-a^6 (b^2-c^2)^2 (b^2+c^2) (b^4
(-5+t) t+c^4 (-5+t) t+b^2 c^2 (-5+7 t-4 t^2))-a^8 (b^4 c^4 (-7+14 t-12
t^2)+b^6 c^2 (3-9 t+4 t^2)+b^2 c^6 (3-9 t+4 t^2))-a^4 (b^2-c^2)^2 (2 b^8
t (2+t)+2 c^8 t (2+t)+b^6 c^2 (2+5 t-3 t^2)+b^2 c^6 (2+5 t-3 t^2)+b^4
c^4 (5-6 t+6 t^2))+a^10 (b^2+c^2) (b^4 (-5+t) t+c^4 (-5+t) t-b^2 c^2
(1-5 t+8 t^2))+a^12 (2 b^4 t (2+t)+2 c^4 t (2+t)+b^2 c^2 (1+9
t^2)))-(-a^2+b^2+c^2) (a^8 t-2 a^6 (b^2+c^2) t+a^4 b^2 c^2 (-2+3
t)-(b^2-c^2)^2 (b^2 c^2 (-1+t)+b^4 t+c^4 t)+a^2 (b^2+c^2) (b^2 c^2 (1-4
t)+2 b^4 t+2 c^4 t)) ((b^2-c^2) (-a^2 c^2-a^4 t+(b^2-c^2)^2 t) (a^4 (-2
b^4+b^2 c^2 (1-2 t))-a^8 t-(b^2-c^2)^3 (b^2+c^2) t+a^6 (2 c^2 t+b^2
(1+t))+a^2 (b^2-c^2) (-b^2 c^2 (-2+t)+2 c^4 t+b^4 (1+t)))+(-b^2+c^2)
(-a^2 b^2-a^4 t+(b^2-c^2)^2 t) (a^4 (-2 c^4+b^2 c^2 (1-2 t))-a^8
t+(b^2-c^2)^3 (b^2+c^2) t+a^6 (2 b^2 t+c^2 (1+t))-a^2 (b^2-c^2) (-b^2
c^2 (-2+t)+2 b^4 t+c^4 (1+t)))) : ... : ...

Pairs {P=X(i),Q=X(j)} (safe error or omission) , for {i,j}: {3,1650},
{23,4}, {186,2}, {1113,10720}, {1114,10719}, {1583,11321}, {2060,14020},
{7486,16950}, {7512,14006}, {8359,17686}, {10128,8364}, {11343,16896},
{14142,17531}, {16355,14001}, {16383,11335}, {16399,14461}

*** If P=X(2)

Q = a^16-2 a^14 (b^2+c^2)-3 a^12 (b^4-4 b^2 c^2+c^4)-(b^2-c^2)^4
(b^2+c^2)^2 (2 b^4-3 b^2 c^2+2 c^4)+2 a^10 (4 b^6-5 b^4 c^2-5 b^2 c^4+4
c^6)-2 a^6 (b^2-c^2)^2 (5 b^6-2 b^4 c^2-2 b^2 c^4+5 c^6)+a^8 (b^8-15 b^6
c^2+29 b^4 c^4-15 b^2 c^6+c^8)+a^4 (b^2-c^2)^2 (3 b^8+2 b^6 c^2-15 b^4
c^4+2 b^2 c^6+3 c^8)+4 a^2 (b^2-c^2)^2 (b^10-b^8 c^2+2 b^6 c^4+2 b^4
c^6-b^2 c^8+c^10) : ... : ...

with (6 - 9 - 13) - search numbers (2.25482286270048,
1.37935466015752, 1.64503916516741).


*** If P=X(4) then Q is the reflection of X(2) in X(13448) :

Q = a^12-2 a^10 (b^2+c^2)+2 a^8 (b^4+b^2 c^2+c^4)-(b^2-c^2)^4 (2
b^4+b^2 c^2+2 c^4)-2 a^6 (b^6+c^6)+2 a^2 (b^2-c^2)^2 (2 b^6-b^4 c^2-b^2
c^4+2 c^6)-a^4 (b^8-3 b^6 c^2+3 b^4 c^4-3 b^2 c^6+c^8) : ... : ...,

on lines X(i)X(j) for these {i, j}: {2,3}, {94,9140}, {2088,14846}.
(6 - 9 - 13) - search numbers ( -54.1570303678729, -54.8836826772129,
66.6326126589962).


*** If P=X(5)

Q = (b^2-c^2)^2 (-a^2+b^2+c^2) (a^8+a^4 b^2 c^2-2 a^6
(b^2+c^2)-(b^2-c^2)^2 (b^4+c^4)+a^2 (b^2+c^2) (2 b^4-3 b^2 c^2+2 c^4))
(a^10-4 a^8 (b^2+c^2)+2 (b^2-c^2)^4 (b^2+c^2)+a^6 (4 b^4+b^2 c^2+4
c^4)-a^2 (b^2-c^2)^2 (5 b^4+4 b^2 c^2+5 c^4)+a^4 (2 b^6+3 b^4 c^2+3 b^2
c^4+2 c^6))+(a^2-b^2) (a^2-c^2) (a^4-(b^2-c^2)^2) (a^16-4 a^14
(b^2+c^2)-(b^2-c^2)^6 (b^2+c^2)^2-4 a^10 (b^2+c^2) (b^4+b^2 c^2+c^4)+2
a^6 (b^2-c^2)^2 (b^2+c^2) (2 b^4+b^2 c^2+2 c^4)+2 a^2 (b^2-c^2)^4
(b^2+c^2) (2 b^4+b^2 c^2+2 c^4)+2 a^12 (3 b^4+5 b^2 c^2+3 c^4)+a^8 (2
b^6 c^2+5 b^4 c^4+2 b^2 c^6)-a^4 (b^2-c^2)^2 (6 b^8+4 b^6 c^2+5 b^4
c^4+4 b^2 c^6+6 c^8)) : ... : ....

with (6 - 9 - 13) - search numbers (-9.94914860821929,
-10.7924224258757, 15.7042562882303).

*** If P=X(20)

Q = (b^2-c^2)^2 (-a^2+b^2+c^2) (-a^^2 c^2+c^4)-2 a^4 (5 b^6-9 b^4
c^2-9 b^2 c^4+5 c^6))-(a^2-b^2) (a^2-c^2) (a^4-(b^2-c^2)^2) (a^16-34 a^8
b^2 c^2 (b^2-c^2)^2-a^14 (b^2+c^2)+11 a^10 (b^2-c^2)^2
(b^2+c^2)-(b^2-c^2)^6 (b^2+c^2)^2+a^12 (-6 b^4+13 b^2 c^2-6 c^4)+a^2
(b^2-c^2)^4 (b^6-9 b^4 c^2-9 b^2 c^4+c^6)-a^6 (b^2-c^2)^2 (11 b^6-8-7 a^4 b^2 c^2+2 a^6
(b^2+c^2)+(b^2-c^2)^2 (b^4+3 b^2 c^2+c^4)-2 a^2 (b^6-2 b^4 c^2-2 b^2
c^4+c^6)) (-5 a^10+8 a^8 (b^2+c^2)+2 (b^2-c^2)^4 (b^2+c^2)+4 a^6 (b^4-8
b^2 c^2+c^4)+a^2 (b^2-c^2)^2 (b^4+14 b27 b^4
c^2-27 b^2 c^4+11 c^6)+a^4 (b^2-c^2)^2 (6 b^8+5 b^6 c^2-38 b^4 c^4+5 b^2
c^6+6 c^8)) : ... : ...

with (6 - 9 - 13) - search numbers (5.41707665942407,
4.53326635151949, -1.99793991195560).

*** If P=X(30)

Q = a^12-a^10 (b^2+c^2)-5 a^6 (b^2-c^2)^2 (b^2+c^2)+a^8 (b^4-b^2
c^2+c^4)-(b^2-c^2)^4 (2 b^4+3 b^2 c^2+2 c^4)+2 a^4 (b^2-c^2)^2 (2 b^4+5
b^2 c^2+2 c^4)+2 a^2 (b^2-c^2)^2 (b^6-3 b^4 c^2-3 b^2 c^4+c^6) : ... :
....

on lines X(i)X(j) for these {i, j}: {2,3}, {125,9530}, {523,1853},
{1899,2452}, {2972,10714}, {3258,11550}}.
(6 - 9 - 13) - search numbers (7.18058761893295, 6.29212512458911,
-4.02953950539252)

Best regards,
Angel Montesdeoca