Support page to my note "The Lamoen circle" / Darij Grinberg

Proof of Theorem 2 in my note "The Lamoen circle"

Darij Grinberg

In my note "The Lamoen circle", I use a theorem:

  Theorem 2.
  A hexagon, whose opposite sides are respectively parallel,
  and whose main diagonals are of equal length, has a circumcircle.

I was asked to give a proof of this theorem. Here is the proof after
H. Dörrie, Mathematische Miniaturen, Wiesbaden 1969.

Call our hexagon ABCDEF, so that AB || DE, BC || EF and CD || FA and
AD = BE = CF. The quadrilateral ABED is an isosceles trapezium
(although it is self-intersecting); hence it has a circumcircle, and
angle ABE = angle ADE. But parallel sides give angle ABE = angle BED
and angle ADE = angle BAD.

Call   a = angle ABE = angle ADE = angle BED = angle BAD.
Analogously, denote
       b = angle BCF = angle BEF = angle CFE = angle CBE,
and    c = angle CDA = angle CFA = angle DAF = angle DCF.

Now
    360° = angle CFE + angle FED + angle EDC + angle DCF
         = angle CFE + angle BEF + angle BED + angle ADE
                     + angle CDA + angle DCF
         = b + b + a + a + c + c = 2(a+b+c),
and thus a+b+c = 180°, so that
    angle CFE + angle EDC = angle CFE + angle ADE + angle CDA
                          = b + a + c = 180°;
hence the points C, D, E and F lie on one circle. On the other
hand, we know that quadrilateral ABED has a circumcircle;
analogously, the quadrilateral BCFE has a circumcircle; hence,
the points B, C, E and F lie on one circle. Thus, the points
B, C, D, E and F lie on one circle. Analogously, the points A,
B, C, D and E lie on one circle. This proves the Theorem.

Dörrie calls this result "Catalan theorem".

  Darij

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Darij Grinberg