Can I assume there is homothety in this situation?

Question

I have found two similar triangles,the triangles $\triangle ABC$ and $\triangle A_1B_1C_1$. The lines connecting their respective vertices intersect at exactly one point, in other words concurrent. May I assume the intersection point is the center of homothety, which moves $ABC$ to $A_1B_1C_1$?

Answer 1

No, e.g. start with an arbitrary 45-90-45 triangle $ABC$, pick any point $E$ on the circle (to ensure $\angle E = 90$) and then points $F$ and $G$ on $EC$ and $EB$ so that $\Delta ECB$ is also 45-90-45.