Hexagon ABCDEF has sides AB and DE of length 2, sides BC and EF of length 7, and sides CD and AF of length 11, and it is inscribed in a circle. Compute the diameter of the circle.Why is A D ¯ the diameter of the circle?

Question

Hexagon ABCDEF has sides AB and DE of length 2, sides BC and EF of length 7, and sides CD and AF of length 11, and it is inscribed in a circle. Compute the diameter of the circle.Why is             A      D        ¯   the diameter of the circle?

Chloe Barr · Accepted Answer

Step 1According to the Inscribed-Angle Theorem,   ∠      A    B    C    ≅  ∠      D    E    F  , and, according to the Side-Angle-Side Theorem,   △      A    B    C    ≅  △      D    E    F  . In particular,   ∠      B    A    C     ≅  ∠      E    D    F  , and             A      C        ¯    ≅            D      F        ¯  . Again, according to the Inscribed-Angle Theorem,   ∠      A    D    C    ≅  ∠      D    A    F  .Likewise,   ∠      B    A    F    ≅  ∠      C    D    E  , and             B      F        ¯    ≅            C      E        ¯  .Since       m    ∠      B    A    C    +      m    ∠      C    A    D    +      m    ∠      D    A    F    =      m    ∠      B    A    F    =      m    ∠      C    D    E    =      m    ∠      E    D    F    +      m    ∠      F    D    A    +      m    ∠      A    D    C    ,,       m    ∠      C    A    D    =      m    ∠      F    D    A  Step 2According to the Angle-Side-Angle Theorem,   △      A    D    F    ≅  △      D    A    C  , in particular,   ∠      A    F    D    ≅  ∠      D    C    A  . These angles are supplementary, though. Consequently, they are right angles. According to the Inscribed-Angle Theorem,             A      D        ¯   is a diameter of the circle.

Ryland Houston · Accepted Answer

Explanation:The lengths of the edges             A      B      C      D      E      F      A        ¯   go 2, 7, 11, 2, 7, 11 which is cyclic of order 2, so the inscribed hexagon has rotational symmetry if turned through             2      π        2  . It follows immediately that any diagonal drawn between pairs of opposite vertices is a perfect diameter.

Hexagon ABCDEF has sides AB and DE of length 2, sides BC and EF of length 7, and sides CD and AF of length 11, and it is inscribed in a circle. Compute the diameter of the circle.

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