We will need the following result from the paper that I mentioned in the comments "An elementary proof and an extension of a theorem of Steinhaus" by M. Kuczma and M. Kuczma that generalizes the Steinhaus theorem.
Thm:(Kuczma and Kuczma) Let $A,B$ be sets with positive inner Lebesgue measure and let $f(x,y)$ be a real-valued function of class $C^{1}$ in a region $D\supseteq A\times B$ with $f_{x}\neq 0,f_{y}\neq 0$ in $D$. Then the set $f(A\times B)$ contains an interval.
Now, to prove our result we shall assume without loss of generality that $\ell_{1}=\{(x,0)|x\in\mathbb{R}\},\ell_{2}=\{(x,1)|x\in\mathbb{R}\}$ and $\ell_{3}=\{(0,y)|x\in\mathbb{R}\}$. Let $U=\{(x,y)\in\mathbb{R}^{2}|x\neq y\}$. Then $U$ is the union of the two regions $\{(x,y)|x<y\}$ and $\{(x,y)|x>y\}.$ Let $f:U\rightarrow\mathbb{R}$ be the function where if $(x,y)\in U$, then $f(x,y)$ is the unique real number such that the points $(x,0),(y,1),(0,f(x,y))$ all fall on the same line. Then any reasonable high school algebra student should be able to tell you that $f(x,y)=\frac{x}{x-y}$. Therefore $f$ is real-analytic on $U$. Also, any calculus 3 student should also be able to tell you that $f_{x}(x,y)=\frac{-y}{(x-y)^{2}}$ and $f_{y}(x,y)=\frac{x}{(x-y)^{2}}$. Therefore the first partial derivatives of $f$ with respect to $x$ and $y$ do not vanish on the set $U$.
If $E_{1},E_{2}\subseteq\mathbb{R}$ are sets with positive Lebesgue measure, then there are subsets $A_{1}\subseteq E_{1},A_{2}\subseteq E_{2}$ of positive Lebesgue measure such that either $a_{1}<a_{2}$ for $a_{1}\in A_{1},a_{2}\in A_{2}$ or $a_{1}>a_{2}$ for $a_{1}\in A_{1},a_{2}\in A_{2}$. Therefore the set $A_{1}\times A_{2}$ is completely contained in some region in $U$, so $f(A_{1}\times A_{2})$ contains some interval. We therefore conclude that the set $E_{3}$ defined in the problem must contain some interval as well.
