i have some problems in my proof of the following task:
Let $l_1, l_2, l_3\in L(\mathbb{R²})$ be three concurrent lines with the common point $q\in\mathbb{R²}$. Show that there exist lines $h_1,h_2\in L(\mathbb{R²})$ with $q\in h_1,h_2$ so that $r_{l_1}\circ r_{l_2}=r_{l_3}\circ r_{h_1}=r_{h_2}\circ r_{l_3} $ where r is the reflection.
Here is my idea of a possible proof: $r_{l_1}\circ r_{l_2}=r_{l_3}\circ r_{h_1}$ is the same as $r_{l_3}\circ r_{l_1}\circ r_{l_2}=r_{h_1}$. q is fix point of the reflection, so that $r_{l_3}\circ r_{l_1}\circ r_{l_2}(q)=r_{h_1}(q)=q$. So $h_1$ is a line, so that the reflection in $h_1$ has the same fix point. So $q\in h_1$.
So i showed the first equation. But is it the right way to proof the complete equation $r_{l_1}\circ r_{l_2}=r_{l_3}\circ r_{h_1}=r_{h_2}\circ r_{l_3} $ ?
Do you have another idea to proof the equation?