HYACINTHOS 29391

[Antreas P. Hatzipolakis]:

 

Let ABC be a triangle and P a point.

 

Denote:

 

O1, O2, O3 = the circumcenters of PBC, PCA, PAB, resp.

 

Oa, Ob, Oc = the reflections of O1, O2, O3 in BC, CA, AB, resp.

 

Which is the locus of P such that the Poncelet Point of OOaObOc lies on the Euler line of ABC?

 

Part of the locus is the Euler line.

 

Which is that Poncelet point for some P's on the Euler line: P = G, N, ....... ?

 

 

[Ercole Suppa] 

 

 

Hi Antreas, 

 

Denote Q = Q(P) the Poncelet point of OOaObOc. 

If P(x:y;z) (barys) then we have Q = {y+z : x+z : x+y} = complement of P .

 

*** The locus of P such that Q(P) lies on the Euler line of ABC is the EULER LINE.

 

*** Some pairs {P=X(i) ∈ Euler line, Q=X(j)}: {{2,2},{3,5},{4,3},{5,140},{20,4},{21,442},{22,427},{23,858},{24,11585},{25,1368},{26,13371},{27,440},{28,21530},{29,18641},{30,30},{140,3628},{186,2072},{235,16196},{237,21531},{297,441},{376,381},{377,405},{378,15760},{379,30810},{381,549},{382,550},{384,6656},{401,297},{402,15184},{403,10257},{404,4187},{405,8728},{411,6831}} 

 

*** Some points:

 

Q(X(199)) = Euler line intercept of X(11)X(7073)

 

= a^4 b^2+a^3 b^3-a b^5-b^6+a^3 b^2 c+a^2 b^3 c-a b^4 c-b^5 c+a^4 c^2+a^3 b c^2+2 a b^3 c^2+b^4 c^2+a^3 c^3+a^2 b c^3+2 a b^2 c^3+2 b^3 c^3-a b c^4+b^2 c^4-a c^5-b c^5-c^6 : : (barys)

 

= (2 p^2-8 R^2+SW)S^2 + 2 p^2 SB SC+SB SC SW :: (barys)

 

= lies on these lines: {2,3}, {11,7073}, {12,1961}, {58,14873}, {86,8044}, {125,17167}, {495,5311}, {496,17017}, {1211,25688}, {2886,27798}, {8287,18165}, {15523,21682}, {19737,19755}, {20531,21098}, {21243,24220}, {23304,29644}, {23922,32778}, {26481,29657}

 

= complement of X(199)

 

= (6-9-13)  search numbers:  [-2.7898528561896405227, -3.6520130768097734689, 7.4566056225555948229]

 

-------------------------------------------------------------------------------------------

 

 

Q(X(406)) = Euler line intercept of X(8)X(18447)

 

= (a^2-b^2-c^2) (a^5+a^4 b-a b^4-b^5+a^4 c+2 a^3 b c-b^4 c+2 a b^2 c^2+2 b^3 c^2+2 b^2 c^3-a c^4-b c^4-c^5) : : (barys)

 

= R S^2 + (-4 a R^2-4 b R^2-4 c R^2+a SW+b SW+c SW)S -R SB SC : : (barys)

 

= lies on these lines: {2,3}, {8,18447}, {10,1060}, {69,20746}, {123,958}, {222,26955}, {255,21912}, {499,17102}, {942,20266}, {1038,1698}, {1040,3624}, {1062,1125}, {1214,19854}, {1437,1899}, {2968,10527}, {3616,18455}, {3739,6389}, {3767,16716}, {5275,23115}, {5276,22120}, {5277,10316}, {5283,14961}, {5714,28836}, {8227,25915}, {10267,25968}, {15668,18642}, {16589,22401}, {17614,24301}, {18592,31198}

 

= complement of X(406)

 

= (6-9-13)  search numbers:  [2.1237624764201535385, 1.2486400144854290873, 1.7960233289159342557]

 

 

Best regards

Ercole Suppa