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This is higher dimension conjecture of Problem 3845 in Crux Mathematicorum and Theorem 2 in here:

PS: This figure is very nice, this is also generalization of Brianchon’s theorem, The Pascal theorem, the Seven circle theorem...etc..... with higher dimension (These are some very nice theorems in plane Geometry). But I don’t know why some one vote down?

I don’t think this question is not good than:

Does this geometry theorem have a name?

The Eyeball Theorem generalized

But the questions above have many vote up. But why some one vote down my question?

My question: The conjecure as follows true for $2$-sphere. Is the conjecture true for n-sphere $(n>2)$?

Conjecture: Let two circle $(C_1)$, $(C_2)$ on n-sphere $(O)$. Let $1, 2, 3, 4, 5, 6$ be six arbitrary points in $(C_1)$. Let $1'$ be arbitrary point in $(C_2)$. The circle $(121')$ meets $(C_2)$ at point $2'$. The circle $(232')$ meets $(C_2)$ again at $3'$, define points $4', 5', 6'$ similarly. Let $P_1, P_2, \cdots, P_6 $ are the center of circles $i{i+1}{i+1}'i'$ for $i=1, 2, \cdots, 6$. Are three planes $(P_1P_4O)$, $(P_2P_5O)$, $(P_3P_6O)$ share the same line ?

  • Circles $(P_1)=(122'1')$, $(P_4)=(455'4')$ are yellow.

  • Circles $(P_2)=(233'2')$, $(P_5)=(566'5')$ are green.

  • Circles $(P_3)=(344'3')$, $(P_6)=(611'6')$ are blue.

enter image description here

enter image description here

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  • $\begingroup$ This figure is very nice, this is also Brianchon’s theorem with higher dimension. But I don’t known why some one vote down $\endgroup$
    – Đào Thanh Oai
    Commented Jul 28, 2018 at 15:43
  • $\begingroup$ I didn't downvote, but I'm guessing people consider your question too elementary for this site. (Whether the picture looks nice doesn't really affect that.) $\endgroup$
    – Jim Conant
    Commented Jul 28, 2018 at 16:03
  • $\begingroup$ @JimConant I don’t think this question is good but many vote up mathoverflow.net/questions/284458/… $\endgroup$
    – Đào Thanh Oai
    Commented Jul 28, 2018 at 17:18
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    $\begingroup$ On the $n$-sphere with $n > 2$, if two circles meet at a point, then they don't necessarily meet at another point. The question seems to be not well-prepared. I vote to close. $\endgroup$
    – Ivan Izmestiev
    Commented Jul 28, 2018 at 18:18
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    $\begingroup$ Deleting the contents of a question is against site policy. You can delete your question if you have not received an answer with positive net score, but otherwise you can't deface a question. Besides, the person who answered put work into the answer; please have some respect for that, and move on. $\endgroup$
    – Todd Trimble
    Commented Jul 31, 2018 at 3:05

1 Answer 1

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If I correctly understand the construction, everything happens in the $3$-dimensional affine space spanned by the circle $C_1$ and the point $1'$. (In particular, $C_2$ has to lie in this space or the points $2',\dots,6'$ won't exist.) Since everything is also on the surface of an $n$-sphere, we can confine attention to the intersection of the affine $3$-space and the sphere, which is a $2$-sphere. So the general case reduces to the case of the $2$-sphere.

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    $\begingroup$ And for the two-sphere it is obvious: they touch the ellipse with foci at centers of circles $(C_1)$ and $(C_2)$. For the proof your just need to check that perpendicular bisectors to $11'$ and $22'$ forms equal angles with $P_1C_1$ and $P_1C_2$. $\endgroup$
    – Arseniy Akopyan
    Commented Jul 28, 2018 at 13:35
  • $\begingroup$ This figure is very nice, this is also Brianchon’s theorem with higher dimension. But I don’t known why some one vote down $\endgroup$
    – Đào Thanh Oai
    Commented Jul 28, 2018 at 15:43
  • $\begingroup$ @ArseniyAkopyan Thank You very much for your answer. But can You delete your answer ? Because I don't wan this theorem are not welcome at here. I want delete my question. $\endgroup$
    – Đào Thanh Oai
    Commented Jul 30, 2018 at 6:22
  • $\begingroup$ @ArseniyAkopyan This question is very nice with me. There are one hundred corollary, special case of this theorem. But this question is not welcome at here. So I want deleted the question to posed on AMM Please help me delete $\endgroup$
    – Đào Thanh Oai
    Commented Jul 31, 2018 at 2:03

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