§ 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.