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What's the distance between the point $D$ and $H$?

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I tried euler geometry thereom but it doesn't gave me the answer unfortunately

Blue
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meh98
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  • Expressed in terms of a function of $r_1,r_2,r_3$ (the radii of the 3 "big circles") ? – Jean Marie Sep 28 '20 at 13:37
  • What is "Euler Geometry theorem" ? Could you give a reference ? – Jean Marie Sep 28 '20 at 14:39
  • Yes i got only the radius of the 3 big blue circles – meh98 Sep 28 '20 at 14:48
  • https://en.wikipedia.org/wiki/Euler%27s_theorem_in_geometry – meh98 Sep 28 '20 at 14:48
  • Thanks. As the circumscribed circle doesn't play a role, this theorem is hopefuly not the good tool. I think there is a solution using complex geometry with formulas one can find here (see formulas (1) and (2) there). – Jean Marie Sep 28 '20 at 14:57
  • I checked out , it is Descartes's theorem , which allow us to find the radius of the little green circle – meh98 Sep 28 '20 at 15:07
  • Also we can find the radius of the black circle too , my goal is to find the distance between the center of the black circle and the center of the green circle – meh98 Sep 28 '20 at 15:13
  • This is Problem 727 of Project Euler. –  Sep 28 '20 at 15:24
  • Yes that is true , i would like to get a formula for this problem , or a starting point to decode it – meh98 Sep 28 '20 at 15:38
  • I have had a look at this very recently posted project which is larger than the posted question. Nevertheless, it would have been correct to mention it into your question. Besides, you haven't yet included in your question that you are looking for a distance formula in terms of $r_1,r_2,r_3$ (my remark). – Jean Marie Sep 29 '20 at 04:24

1 Answers1

2

If all you're looking for is a formula, here goes.

For a triangle $\triangle ABC$, let $I$=incenter, $Ge$=Gergonne point, $L$=de Longchamps point, $S$=inner Soddy point (called $D$ in the OP. It's triangle center $X_{176}$, also called the equal detour point. See Dergiades below.)

Dergiades' paper The Soddy Circles gives a formula at the end of section 3.1 for $S$ in terms of the distance between $I$ and $Ge$.

A formula for $GeI$ in terms of $IL$ can be found towards the end of the Mathworld Gergonne Point entry.

A formula for $LI$ is given in the Mathworld de Longchamps Point entry.

Beware of typos.

Hope this helps.

brainjam
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