COURSE OF PROJECTIVE GEOMETRY


§ 11: answers


O43 From the Steiner theorem it follows that l.p is mapped to l.m, and l.m to m.p. So l.m is invariant if and only if l, m and p go through one point.

O44

We find X ' when we connect A with XA'.p and intersect the connection line with m.
This only fails when {A, A'} and {l.p, l.m, m.p} have a point in common. Check that we can't do the construction of X ' in this last case.

O45
Let l1 be the connection line of a.b ' and a '.b, and l2 the connection line of a.c ' and a '.c. Then l1.l2 is the required Pappus point. The connection line of l.a ' and a.φ(l) also goes through this point.
So connect l.a ' with the Pappus point and intersect the connection line with met a. Then you find a.φ(l) as intersection point. So φ(l) is the connection line of M and this intersection point.

O46
(S, A1, D, A2) =R (S, B1, G, B2) =P (S, C1, F, C2).

O47
Hint: if necessary, first project A, B and C from a help point onto a help line.


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