Take a point on the straight line B 1D as PO 1, PO2 and PO3 respectively.

Vertical to B 1d 1, B 1c, b1a.

In O 1, O2, O3

Then PO 1⊥ faces a1c1,and PO2 ⊥ faces b1C.

PO3⊥ plane A 1B,

O 1，O2，O3

They are o 1n ⊥ a 1d 1, o2m ⊥ cc 1, o3q ⊥ ab.

The vertical feet are m, n, q,

Even PM, PN and PQ can be obtained from the three vertical theorems, PN⊥A 1D 1.

PM⊥CC 1

; PQ⊥AB，

Because all the faces in the cube are congruent, P0 1=PO2=PO3, ∴PM=PN=PQ, that is, the distance from P to the straight line of three sides AB, CC 1, A 1D 1. Are all equal, so there are infinite points to meet the conditions, so choose D.