HYACINTHOS 27033

Kadir Altintas] (*)

 

Let ABC be a triangle and A'B'C' the cevian triangle of G

Denote:

A circle (A', x) intersects BC at Bc, Cb near to B, C, resp.

A circle (B', y) intersects CA at Ca, Ac near to C, A, resp.

A circle (C', z) intersects AB at Ab, Ba near to A, B, resp.

 

M1, M2, M3 = the midpoints of AbAc, BcBa, CaCb, resp.

1. ABC, M1M2M3 are perspective.

2. ABC, M1M2M3 share the same centroid

3. Ab, Ac, Bc, Ba, Ca, Cb lie on a conic.

(*) Romantics of Geometry 1520

 

 

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If x, y, z (or a/2 -x =: X, b/2 -y =: Y, c/2 -z =: Z) are symmetric functions of a,b,c ie f(a,b,c), f(b,c,a), f(c,a,b) then the point of concurrence is a triangle center.

Which is it if x, y, z = exradii r_a, r_b, r_c, resp. or altitudes h_a, h_b, h_c, resp. ?

αph

 

 

[Angel Montesdeoca]:


Let ABC be a triangle and MaMbMc the medial triangle.

1.     r_a, r_b, r_c the exradii.
The circle Ma(r_a) of center Ma and radius r_a, cut BC into A' and A".
The points B', B", C', C" are defined cyclically.

 *** M1, M2, M3 = the midpoints of B'C", C'A", A'B", resp.
 
  The triangles ABC, M1M2M3 are perspective, with perspector
  
  U = ( a(b+c-a)/(a(b+c-a)-2S) : ... : ...),
  
  lies on lines X(i)X(j) for these {i, j}:  {2, 13436}, {9, 3084}, {281, 1585}, {346, 13458};
 and (6 - 9 - 13) - search numbers (10.8889191336017, -1.96209190005363, -0.0266191874100545).
 
  ***  N1, N2, N3 = the midpoints of B"C', C"A', A"B', resp.
 
 The triangles ABC, N1N2N3 are perspective, with perspector
  
  V = ( a(b+c-a)/(a(b+c-a)+2S) : ... : ...),
  
  lies on lines X(i)X(j) for these {i, j}:  {2, 13453}, {9, 3083}, {281, 1586}, {346, 13425};
 and (6 - 9 - 13) - search numbers (3.29230700976681, 2.12455169577637, 0.650294688016054).
 
 *** The triangles M1M2M3,  N1N2N3 are perspective, with perspector X(6666).
 
2.  h_a, h_b, h_c are the  altitudes.
  
  The circle Ma(h_a) of center Ma and radius h_a, cut BC into A' and A".
The points B', B", C', C" are defined cyclically.

 *** M1, M2, M3 = the midpoints of B'C", C'A", A'B", resp.
 
  The triangles ABC, M1M2M3 are perspective, with perspector  X(589).
  
   ***  N1, N2, N3 = the midpoints of B"C', C"A', A"B', resp.
 
 The triangles ABC, N1N2N3 are perspective, with perspector X(588).
 
  *** The triangles M1M2M3,  N1N2N3 are perspective, with perspector  X(3589).
  
   More details in: http://amontes.webs.ull.es/otrashtm/HGT2018.htm#HG100118
 

  Angel Montesdeoca