Consider the birational map $\chi$ given by the blow up of the points $(1:0:0)$, $(0:1:0)$, and $(0:0:1)$ of $\mathbb P^2$, followed by the contraction of the strict transforms of the three lines passing through two of the above points. I would like to show that the proper transform by $\chi$ of the linear system of lines is the linear system of conics passing through $(1:0:0)$, $(0:1:0)$, and $(0:0:1)$.
Thank you in advance!
Edit : If possible without writing the maps explicitly in coordinates.
A generic line $\ell$ cuts each of the three contracted lines in one point. Thus the proper transform $L$ of it passes through the contraction points. It is smooth on the contractions points because it cuts each of the contracted lines transversally and just one time. Thus $L$ is smooth everywhere.
In order to compute its degree it suffices to remark that it intersects each the proper transform of the exceptional divisors of the first blow up (which is a line) only in two points. Thus $L$ is a conic (Bezout).
I hope that this is conceptual enough.
Otherwise, use intersection theory on surfaces and you will obtain the same result.