# Miquel circles on sphere

Consider a sphere $\Bbb{S}$ on Euclide 3D space.

We well-known that a "line" connecting two points $X$ and $Y$ on $\Bbb{S}$ is the great circle of $\Bbb{S}$ which passes through points $X$ and $Y.$

In 3D space, we also denote by $(X,Y,Z)$ the circle passing through three points $X,$ $Y$ and $Z.$ Consider a triangle $ABC$ on a sphere $\Bbb{S}.$ Points $D,$ $E$ and $F$ are arbitrary on lines $BC,$ $CA$ and $AB$ respectively.

We consider the circles $(A,E,F),$ $(B,F,D),$ $(C,D,E).$

Question 1. Why can the circles $(B,F,D),$ $(C,D,E)$ meet again at $X$?

Define similarly, the intersecions $Y$ and $Z.$

I see by geogebra

Question 2. The "lines" (on sphere) $DX,$ $EY$ and $FZ$ are concurrent. Could you please give a solution?

Question 3. Is this problem true on $n-$ sphere embedded in an $(n + 1)-$ dimensional Euclidean space?

Question 4. If we use orthogonal projection to project this configuration on a plane, then what is plane geometry problem which we obtain?

• Look at the linear functions that a constant on your circles; note that the lines DX, EY and FZ are zero sets of their linear combinations, and think. – Anton Petrunin Jul 5 '18 at 13:01
• You can reference here: mathoverflow.net/questions/273968/… if You let Ellipsoid is a ball this is Miquel of six circle on a ball. I think Miquels in a triangle is special case. Professor Fedor Petrov proved the result very simple – Đào Thanh Oai Jul 6 '18 at 7:34
• You can see reference here: Question B5≡B1 or B5≠B1? if You let two conic are two circles, Ellipsoid is a ball this is Miquel of six circle on a ball. I think Miquels in a triangle is special case. Professor Fedor Petrov proved the result very simple – Đào Thanh Oai Jul 6 '18 at 7:49