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3D analytic geometry (skew lines)

Source: Romanian TST 1978, Day 3, P3

September 30, 2018
analytic geometrygeometryskew lines3D geometry3D analytic geometry

Problem Statement

a) Let D1,D2,D3 D_1,D_2,D_3 be pairwise skew lines. Through every point P2∈D2 P_2\in D_2 there is an unique common secant of these three lines that intersect D1 D_1 at P1 P_1 and D3 D_3 at P3. P_3. Let coordinate systems be introduced on D2 D_2 and D3 D_3 having as origin O2, O_2, respectively, O3. O_3. Find a relation between the coordinates of P2 P_2 and P3. P_3.
b) Show that there exist four pairwise skew lines with exactly two common secants. Also find examples with exactly one and with no common secants.
c) Let F1,F2,F3,F4 F_1,F_2,F_3,F_4 be any four secants of D1,D2,D3. D_1,D_2, D_3. Prove that F1,F2,F3,F4 F_1,F_2, F_3, F_4 have infinitely many common secants.
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