Pair of elliptic involution on a projective line

Question

In a projective plane $\alpha$ let $u$ and $v$ be two lines and $\Theta$, $\Psi$ be elliptic involution on $u$, respectively, $v$. Prove that there exists two pencil of rays on which $\Theta$ and $\Psi$ induces the same involution.

Answer 1

Let $M$ be the intersection of $u$ and $v$, and let $w$ be the line $\Theta(M)\Phi(M)$. Let $N$ be a fixed arbitrary point of $u$ and $N'=\Theta(N)$. Let $\pi_N$ be the perspectivity with center $N$ from $w$ to $v$ and $\pi_{N'}$ be the perspectivity with center $N'$ from $v$ to $w$. Then consider the projectivity $$\pi_{N'}\circ \Phi\circ \pi_{N}.$$ This projectivity has two fixed points. We prove that for any $F$ of these fixed points $\Theta$ and $\Phi$ determine the same involution on the pencil through $F$; i.e., $$\pi^{-1}_F\circ \Phi\circ \pi_F = \Theta,$$ where $\pi_F$ is the perspecitvity from $u$ to $w$ through $F$.

Clearly, $\pi^{-1}_F\circ \Phi\circ \pi_F$ is an involution. So it is enough to prove that it maps two points to the same points as $\Theta$.

It maps $M$ to $\Phi(M)$, since $\pi_F$ sends $M$ to $M$, $\Phi$ sends $M$ to $\Theta(M)$ and $\pi^{-1}_F$ sends $\Theta(M)$ to $\Phi(M)$.

$F$ is a fixed point of $\pi_{N'}\circ \Phi\circ \pi_{N}$, which means that $\Phi$ sends $FN\cap v$ to $FN'\cap v$. Thus $\pi_F$ sends $N$ to $FN\cap v$, $\Phi$ sends it to $FN'\cap v$, and $\pi^{-1}_F$ sends this point to $N'$. Thus $\pi^{-1}_F\circ \Phi\circ \pi_F$ sends $N$ to $N'$, such as $\Theta$.