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I have been struggling a bit with a math-task.

The question is as follows: The hexagon ABCDEF is inscribed in a circle. The length of AF is 31 while the other five sides has the length of 81. Tell the sums of the diagonals from A.

Dividing the hexagon into a quadrilateral AFEB and using Ptolemaios theorem gives $AE\cdot BF=AF\cdot BE+AB\cdot EF=31\cdot BE+81^2$

Dividing the hexagon into a quadrilateral AFED and using Ptolemaios theorem gives $AE\cdot DF=AF\cdot DE+AD\cdot EF=31\cdot81+81\cdot AD=81(31+AD)$

I just keep getting more unknown variables and I really don't know how to proceed. Please help, I hope my effort has been explained clearly enough. Also, please give tips on how to structure and declare my questions and explainations better.

Mersault
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1 Answers1

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$\hspace1in$ A hexagon in a circle

Label the sides and diagonals of the hexagon as shown.

  • $a = AF = 31$
  • $b = AB = BC = CD = DE = EF = 81$
  • $c = AC = BD = DF$
  • $d = AD = CF$
  • $e = AE$.

Apply Ptolemy's theorem to quadrilaterals $ABCD$, $ACDF$ and $ADEF$, we obtain

$$\begin{align} c^2 &= bd + b^2\\ d^2 &= c^2 + ab\\ ce &= bd + ab \end{align}$$

The first two equations tell us

$$d^2 = b(a+b+d) \iff d^2 = 81(d+112) \iff (d-144)(d+63) = 0 \implies d = 144$$ Substitute this back into first equation, we obtain

$$c = \sqrt{b(b+d)} = \sqrt{81(81+144)} = 135$$

Throw this into last equation, we find $$e = \frac{b(a+d)}{c} = \frac{81(31+144)}{135} = 105$$

The desired sum of diagonals from $A$ equals to

$$AC + AD + AE = c + d + e = 135 + 144 + 105 = 384$$

achille hui
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