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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

The Lamoen circle

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*Darij Grinberg*