ADGEOM 4856 * ADGEOM 4859

#4856

Dear geometers,

 

I call line X(3)X(49) of ABC by Sine-triple-angle axis of ABC.

 

Let ABC be a triangle.

 

Orthic triangle A'B'C'.

 

Let da be the reflection of Sine-triple-angle axis of AB'C' in line AA'.

 

Define similarly the line db and dc.

 

Then the lines da,db,dc are concurrent.

 

Which is this point?

 

Best regards,

Tran Quang Hung.

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#4859

[Tran Quang Hung]:

I call line X(3)X(49) of ABC by Sine-triple-angle axis of ABC.

Let ABC be a triangle.
Orthic triangle A'B'C'.
Let da be the reflection of Sine-triple-angle axis of AB'C' in line
AA'.
Define similarly the line db and dc.

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

*** The barycentric equation of the reflection of Sine-triple-angle
axis of AB'C' in line AA' is

da: a^2 (b^2-c^2) (a^6-a^2 (b^2-c^2)^2-a^4 (b^2+c^2)+(b^2-c^2)^2
(b^2+c^2)) x+(-2 a^10+2 (b^2-c^2)^5-a^2 (b^2-c^2)^3 (3 b^2+c^2)+a^8 (5
b^2+c^2)+a^6 (-3 b^4-2 b^2 c^2+c^4)+a^4 (b^6+3 b^4 c^2-5 b^2 c^4+c^6))
y+(2 a^10+2 (b^2-c^2)^5-a^2 (b^2-c^2)^3 (b^2+3 c^2)-a^8 (b^2+5 c^2)+a^6
(-b^4+2 b^2 c^2+3 c^4)-a^4 (b^6-5 b^4 c^2+3 b^2 c^4+c^6)) z=0.

The lines da,db,dc are concurrent at D = midpoint of X(4)X(5962)

D = 2 a^16
-7 a^14 (b^2+c^2)
+4 a^12 (2 b^4+5 b^2 c^2+2 c^4)
-a^10 (3 b^6+17 b^4 c^2+17 b^2 c^4+3 c^6)
-2 a^8 b^2 c^2 (b^4-10 b^2 c^2+c^4)
+3 a^6 (b^2-c^2)^2 (b^2+c^2)^3
-4 a^4 (b^2-c^2)^2 (2 b^8-b^6 c^2+2 b^4 c^4-b^2 c^6+2 c^8)
+a^2 (b^2-c^2)^4 (7 b^6+b^4 c^2+b^2 c^4+7 c^6)
-2 (b^2-c^2)^6 (b^4+b^2 c^2+c^4) : .... : ....


D is the midpoint of X(4) and X(5962)
D is the reflection of X(12095) in X(5)
D lies on lines X(i)X(j) for these {i, j}: {4,52}, {5,12095},
{30,131}, {381,13557}, {403,14769}, {924,13851}, {5203,9880},
{7547,14889}}

(6 - 9 - 13) - search numbers of D: (-1.77451732881372,
-1.23193647064380, 5.31255157488257)

Angel Montesdeoca