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I was trying to draw the eight point circle for a convex quadrilateral. The only condition that I know of is that it must be a convex quadrilateral with perpendicular diagonals.

This is a picture of a successful construction using GeoGebra. I constructed it by first constructing perpendicular diagonals (that is BG and FH) then completing the quadrilateral from there on.

8-point circle

Even if we vary the length of line segment EH, the diagonals will still be perpendicular (as all angles at E are perpendicular), and taking care it remains a convex quadrilateral, I arrived at the following:

putative 8-point circle, but with two points missing

Where did the other few points go? Am I missing some other conditions?

Cheng
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    If you extend $GH$ and $BH$, you will get the other two points you want. – justadzr Jan 16 '21 at 04:18
  • @Yourong 'DZR' Zang Oh wait, so the definition of eight point circle includes the case where the lines need to be extended? Oops. Thanks! – Cheng Jan 16 '21 at 04:22
  • The point is that sometimes you cannot get the feets of the perpendiculars if you do not extend the lines – justadzr Jan 16 '21 at 04:24

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