Equilateral simplices in normed 4-space

Let $E$ be a 4-dimensional real normed space, $x\ge3/4$ a positive number, and $P\subset E$ a 3-plane. It is proved that there exist 4 equidistant points $A_1$, $A_2$, $A_3$, $A_4\in P$ and a point $A_5\in E$ such that $\|A_5A_i\|=x\cdot\|A_1A_2\|$ for $i=1,2,3,4$. In particular, $E$ contains an equilateral simplex.