domingo, 15 de enero de 2017

1414. Real or imaginary?

    A problem:

    If the three green triangles  ABF ,  BCD  and  CAE  are similar, then... Have the triangles  ABC  and  DEF  the same barycenter  G ?
    Pepe Chapuzas imagined that these real triangles were imaginary, so he could work with complex numbers...

    Dear Teacher:
    Let's consider the complex numbers a, b, c, d, e, f, g associated with the points A, B, C, D, E, F, G... Since the green triangles are similar (same angles, proportional sides), then...

(db) / (cb)  =  (ec) / (ac)  =  (fa) / (ba)  =  h
    Thusly
d  =  ch − bh + b
e  =  ah − ch + c
f  =  bh − ah + a
    Hence
(d+e+f) / 3  =  (a+b+c) / 3  =  g

    Another problem:
    In a circle of radius  r  we have three equidistant chords (red), and the distance between any two chords is  r  too, then... Are the midpoints of the chords the vertices of an equilateral triangle?
    Pepe Chapuzas solved it too...

    Dear Teacher:
    I have a circle... I can assume that its center is  0  and its radius is  2  to simplify, can I not? And I can assume that the endpoints of the chords are... (in polar coordinates):

α   and   2 β−60°
β   and   2 γ−60°
γ   and   2 α−60°
    The midpoints are...
a  =  1 α + 1 β−60°  =  α + 1 β · 1 300°
b  =  1 β + 1 γ−60°  =  β + 1 γ · 1 300°
c  =  1 γ + 1 α−60°  =  γ + 1 α · 1 300°
    Let me calculate...
(ab) / (cb)  =
=  (α + 1 β · 1 300° − β − 1 γ · 1 300°) / (γ + 1 α · 1 300° − β − 1 γ · 1 300°)  =
=  (α + 1 β · 1 240° + 1 γ · 1 120°(1 α · 1 300° + β · 1 180° + 1 γ · 1 60°)  =
=  1 60°

    If  A ,  B ,  C  are the corresponding real points to  a ,  b ,  c , then the sides  BA  and  BC  have the same length and the angle between them measures 60°, that is, the triangle ABC is equilateral.

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