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The Three Tangents Theorem
Suppose that C is a conic. If A,B,C are p-points such that
AB touches C at P,
BC touches C at R and CA touches C at Q,
then AR, BQ and CP are concurrent.
Proof
By the Three Point Theorem, there is a projective transformation t
which maps C to C0 and P,Q,R to X,Y,Z respectively. Since t preserves
tangency, it maps A to A', the intersection of the tangents at X and Y.
The tangent at a point can be obtained from the Joachismsthal Formula
but in this case, it is easy to verify that the tangent at X is L : y+z = 0:
C0 has equation xy+yz+zx = 0, so it meets L where yz = 0. Then we
must have y = 0 or z = 0. We must also have y+z = 0, so y = z = 0, so
that L meets the conic at the single point X, i.e. is the tangent at X.
By symmetry, the tangent at Y is M : x+z = 0. By inspection, L and M
meet where x = y = -z, so that A' is [1,1,-1]. Observe that Z and A' lie
on the p-line x = y, so ZA' has equation x = y.
By the algebraic symmetry, YB' is x = z, and XC' is y = z. The p-point
U[1,1,1] lies on ZA',YB' and XC', so the p-lines are concurrent at this
p-point.
Applying t-1, we see that AR, BQ and CP are concurrent.
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