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?