Determine whether the following construction is possible:

Dissect an equilateral triangle into 9 similar triangles with angles of 45, 60 and 75 degrees; one at each corner and the other six meeting at a point and forming a hexagon.

Inscribe 6 points (A,X,B,Y,C,Z in order) around a circle with centre O.

                              X                        B

                                             l               
                                 r                       
                                                           s

              A               x         O                          Y


                             n                            m      

                                              t
                             Z                           C


Join  A, B & C to form an equilateral triangle.
Inscribe point "n" on AC as an intersection with XZ.
With centre A and radius |An| inscribe point "x" on AO.  [This creates a 30º iscoseles triangle].

Extend nx to intersect AB at r.
Join O to n and O to r.

Repeat for vertices B and C.

Now while this construction does have the triangles at the vertices of the equilateral having 45º, 60º and 75º, those around the incentre do not comply. 

Now let me see what broll actually said.


Looking at that drawing and offsetting line nr closer and closer to A so that angle nrO becomes 60º, the other angles of the ΔnrO are more like 71º and 49º (though not exactly).

Edited on July 20, 2012, 10:25 pm

Posted by brianjn on 2012-07-20 08:54:59