Problem of the Day: June 26

All six sides of the convex hexagon ABCDEF are equal. In addition, AD = BE = CF. Prove that a circle can be inscribed in the hexagon.

Comments

  1. UnknownJuly 1, 2019 at 10:19 AM

    My quick sketch of a proof, which hopefully is correct:

    (1) Use SSS congruence to show that triangles ACF and CAD are congruent, which (along with BAC being isosceles) means that angles A and C of the hexagon are congruent. From symmetry, it follows that C and E are also congruent. So, angles A, C, and E of the hexagon are congruent.

    (2) Use SAS congruence to show that BD=DF=FB, which means that triangle BDF is equilateral. Similarly, triangle ACE is also equilateral.

    (3) Use symmetry to show that the incenter of BDF is equidistance from the 6 sides of hexagon ABCDEF.

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