`P_1n dP_2` are planes passing through origin `L_1a n dL_2` are two lines on`P_1a n dP_2,` respectively, such that their intersection is the origin. Show that there exist points `A ,Ba n dC ,` whose permutation `A^(prime),B^(prime)a n dC^(prime),` respectively, can be chosen such that `A` is on `L_1,BonP_1` but not on `L_1a n dC` not on `P_1;` `A '` is on `L_2,B 'onP_2` but not on `L_2a n dC '` not on `P_2dot`

Explanation

Fig A 2.29 shows the possi9ble sitution for planes`P_(1) and P_(2)` and the lines `L_(1) and L_(2)` :
Now if we choss points A,B and C as A on `L_(1)` B on th3e line of interection of ` P_(1) and P_(2)` but other than th3 origin and C on `L_(2)` again other than the origin. then we can consider.
a correspondes to one A', B', C'
B corresponds to one of the remaining of A' B' and C'
C corresponds to thrid of A', B', and C', e.g.
A'= C' B = , C' =A
Hence one permutation of [ A B C ] is [ C B A] . hence proved.