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enter image description here

enter image description here

but how to deduce it ? Thank you .

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  • $\begingroup$ For the record, this is the forum for the software Mathematica, you were probably looking for math.stackexchange.com $\endgroup$ – Feyre Aug 4 '16 at 9:49
  • $\begingroup$ Welcome to Mathematica.SE! I suggest the following: 1) As you receive help, try to give it too, by answering questions in your area of expertise. 2) Take the tour! 3) When you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge. Also, please remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign! $\endgroup$ – Michael E2 Aug 4 '16 at 10:51
  • $\begingroup$ See en.wikipedia.org/wiki/Envelope_(mathematics) $\endgroup$ – Michael E2 Aug 4 '16 at 11:03
  • $\begingroup$ I don't think your lines on above illustrations are actually equal-length, since they seem to be from $(0, a)$ to $(1-a, 0)$. $\endgroup$ – kirma Aug 10 '16 at 7:43
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It's an astroid and not a circle. In any event:

With[{r = 1, n = 30}, 
     Graphics[{AbsoluteThickness[1], 
               Table[Line[r {{Cos[θ], 0}, {0, Sin[θ]}}], {θ, 0, π/2, π/(2 n)}]}]]

astroid as envelope

A short proof that the astroid is the correct envelope:

{x, InterpolatingPolynomial[{{Cos[θ], 0}, {0, Sin[θ]}}, x]} /.
    First[Solve[D[InterpolatingPolynomial[{{Cos[θ], 0}, {0, Sin[θ]}}, x], θ] == 0, x]]
    // FullSimplify
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  • $\begingroup$ Fair enough, I'm too trusting with what people claim to know. $\endgroup$ – Feyre Aug 4 '16 at 9:48
  • $\begingroup$ great! Thank you! $\endgroup$ – DMYZK Aug 4 '16 at 11:30
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Based on your image, but not your statement on lengths of lines:

With[{eqn = a - a x / (1 - a)},
 Show[
  Quiet@Plot[Table[eqn, {a, 0, 1, 1 / 20}], {x, 0, 1},
   Evaluated -> True, AspectRatio -> Automatic, PlotRange -> {0, 1},
   PlotStyle -> Gray],
  Plot[MaxValue[{eqn, 0 <= a <= 1}, a], {x, 0, 1},
   Evaluated -> True, PlotStyle -> Directive[Red, Thick]]]]

enter image description here

For the intended plot your can replace eqn with the following:

eqn = Sin[Pi a / 2] - Tan[Pi a / 2] x

enter image description here

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  • 2
    $\begingroup$ Solve[{#, D[#, a]} &[x/a + y/(1 - a) == 1], {x, y}, {a}] also yields the red parabola (1 + (x - y)^2 - 2 (x + y)). $\endgroup$ – Michael E2 Aug 10 '16 at 19:00
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$$\sqrt{1-x^{2/3}}-\sqrt{1-x^{2/3}}x^{2/3}$$

Plot[Sqrt[1 - x^(2/3)] - Sqrt[1 - x^(2/3)] x^(2/3), {x, 0, 1}, 
AspectRatio -> 1, PlotStyle -> {Thick, Red, Dashed}

enter image description here

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