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I am very curious, if it is possible to invert the Japanese Theorem. So, if the middlepoints of the inner circle create a rectangle, is the quadrilateral on the outside of the four circles always a cyclic quadrilaterals?

I searched for hours in the web but found nothing. And after trying a lot of different ideas to proof it, I want to ask for help now. I am convinced, that it is invertable, because I wasn't able to find a exeption by using GeoGebra. Have anyboy an idea, how to solve this problem?

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  • $\begingroup$ Can you please first state this "Japanese theorem"? I think many users such as myself have not heard of it. Also, by "invert the Japanese Theorem" are you asking if the converse of the theorem is true, or something else? $\endgroup$ – YiFan Nov 23 '18 at 0:12
  • $\begingroup$ @YiFan See my question where I recall what the Japanese theorem is and an ill-known pretty extension ; math.stackexchange.com/q/2614053 $\endgroup$ – Jean Marie Nov 23 '18 at 0:43
  • $\begingroup$ @YiFan The Japanese Theorem told us, that the four midpoints of the inner circle of the by the diagonals seperated triangles in an cyclic quadrilateral always are the four corners of a rectangle. So, by invert it, I'd like to know, if the four midpoints of the inner circle are the four corners of a rectangle, is the outer quadrilateral always a cyclic quadrilateral? $\endgroup$ – Phoenix Smaug Nov 23 '18 at 11:50
  • $\begingroup$ See this one with a drwawing en.wikipedia.org/wiki/… $\endgroup$ – Moti Nov 23 '18 at 20:39
  • $\begingroup$ @YiFan but with all the information the answer was not given $\endgroup$ – Moti Nov 23 '18 at 20:49
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I do not think it works - if this what you mean

enter image description here

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