Mathematics

Problem

Let

A(1,0,0)

B(0,1,0)

C(0,0,1)

and

D(t,t,t)

where

t>0

. (i) Obtain an expression for the square of the distance between

D

and each of

A

B

and

C

in terms of

t

. (ii) Determine the range of

t

for which

D

is nearest to

A

among the three segments

DA

DB

and

DC

. (iii) Taking

t=1

, compute all six pairwise distances among

A

B

C

and

D

, and list them in ascending order, clearly indicating any equal lengths.