ADGEOM 4866 * ADGEOM 4867

#4866

Dear geometers,

 

Let ABC be a triangle with orthocenter H.

 

P=X(110) of ABC.

 

da be the reflection of Euler line of PBC in line HA.

 

Define similarly, the lines db and dc.

 

Then da,db,dc are concurrent.

 

Which is this point?

 

Best regards,

Tran Quang Hung.

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#4867

[ Tran Quang Hung]:

Let ABC be a triangle with orthocenter H.
P=X(110) of ABC.
da be the reflection of Euler line of PBC in line HA.

Define similarly, the lines db and dc.

Then da,db,dc are concurrent.
Which is this point?

*** da,db,dc are concurrent at W = midpoint of X(265) and X(14989)

W = 4 a^16
-9 a^14 (b^2+c^2)
+a^12 (-6 b^4+44 b^2 c^2-6 c^4)
+14 a^10 (2 b^6-3 b^4 c^2-3 b^2 c^4+2 c^6)
-3 a^8 (5 b^8+15 b^6 c^2-44 b^4 c^4+15 b^2 c^6+5 c^8)
-a^6 (9 b^10-71 b^8 c^2+63 b^6 c^4+63 b^4 c^6-71 b^2 c^8+9 c^10)
+a^4 (b^2-c^2)^2 (4 b^8+14 b^6 c^2-63 b^4 c^4+14 b^2 c^6+4 c^8)
+6 a^2 (b^2-c^2)^4 (b^6-2 b^4 c^2-2 b^2 c^4+c^6)
-(b^2-c^2)^6 (3 b^4+7 b^2 c^2+3 c^4) : ... : ....



W is the midpoint of X(265) and X(14989).
W is the reflection of X(i) in X(j), for these {i, j}: {14677,12079},
{14934,546}, {16340,10113}.
W lies on lines X(i)X(j) for these {i, j}: {30,74}, {523,3627},
{546,14934}, {10113,16340}, {12295,16168}.
(6 - 9 - 13) - search numbers of W: (-12.3731515365849,
-12.2736427038067, 17.8484870629667).

Angel Montesdeoca