ADGEOM 5177 * ADGEOM 5182

#5177

Let ABC be a triangle with centroid G. A'B'C' is pedal triangle of G.

 

Then reflections of Brocad axis of triangles GBC, GCA, GAB in GA', GB', GB' respectively are concurrent.

 

Which is this concurrent point?

 

Best regards,

Tran Quang Hung.

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#5182

[Tran Quang Hung]

 

Let ABC be a triangle with centroid G. A'B'C' is pedal triangle of G.

 

Then reflections of Brocad axis of triangles GBC, GCA, GAB in GA', GB', GB' respectively are concurrent.

 

Which is this concurrent point?

 

Best regards,

Tran Quang Hung.

 

 

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[Ercole Suppa]

 

 

Then reflections of Brocad axis of triangles GBC, GCA, GAB in GA', GB', GB' respectively concur at point:

 

 

X = 77 a^10-205 a^8 b^2+101 a^6 b^4+116 a^4 b^6-124 a^2 b^8+35 b^10-205 a^8 c^2+107 a^6 b^2 c^2+90 a^4 b^4 c^2+47 a^2 b^6 c^2-91 b^8 c^2+101 a^6 c^4+90 a^4 b^2 c^4+90 a^2 b^4 c^4+56 b^6 c^4+116 a^4 c^6+47 a^2 b^2 c^6+56 b^4 c^6-124 a^2 c^8-91 b^2 c^8+35 c^10 :: (barys)

 

= (1188 R^2-27 SB-27 SC-414 SW)S^4 + (-324 R^2 SB SC+162 SB SC SW-9 SB SW^2-9 SC SW^2+16 SW^3)S^2 +12 SB SC SW^3 :: (barys)

 

= on lines X(i)X(j) for these {i,j}: {15694,24206}

 

= (6-9-13) search numbers [0.887705658890037360, -0.587071572915610678, 3.63738834366901541]

 

 

Best regards

Ercole Suppa