Let $A_0A_1...A_n$ be a simplex in $\Bbb E^n.$ Let $B_{ij}$ be a point on the edge $A_iA_j,\ 0\le i\not=j\le n.$

Denote by $\beta_i$ the hyperplane passing through the points $B_{i0},$ $B_{i1},$ $B_{ii-1},$ $B_{ii+1},$ $...,$ $B_{in}.$

Assume that the reflection of $A_i$ in the hyperplane $\beta_i$ lies on the hyperplane $A_0A_1...A_{i-i}A_{i+1}...A_n$ for all $i$.

My question.Can we show that $B_{ij}$ must be midpoints of the edges $A_iA_j?$