Conic sections are extensively encountered in a wide range of disciplines, including optics, physics, and various other fields. Consequently, the geometric algebra community is actively engaged in developing frameworks that enable efficient support and manipulation of conic sections. Conic-conic intersection objects are known and supported by algebras specialized in conic sections representation, but there is yet no elegant formula to extract the intersection points from them. This paper proposes a method for point extraction from an conic intersection through the concept of pencils. It will be based on QC2GA, the 2D version of QCGA (Quadric Conformal Geometric Algebra), that we also prove to be equivalent to GAC (Geometric Algebra for Conics).