Embedding products of trees into higher rank

We show that there exists a quasi-isometric embedding of the product of n copies of HR² into any symmetric space of non-compact type of rank n, and there exists a bi-Lipschitz embedding of the product of n copies of the $3$-regular tree T₃ into any thick Euclidean building of rank n with co-compact affine Weyl group. This extends a previous result of Fisher--Whyte. The proof is purely geometrical, and the result also applies to the non Bruhat--Tits buildings.