In this project, we defined a mid-tangent point with respect to a fixed point X and a tangent at a point Y on a planar curve C as a point on the tangent that is equidistant from X and Y. We studied the locus of mid-tangent points of conic sections. We found that the locus of mid-tangent points of most conic sections are non-linear curves. However, we observed and proved by using Euclidean geometry that the locus of mid-tangent points of circles are straight lines. The mapping defined by mid-tangent points was studied further. The similarity between a mid-tangent mapping and a stereographic projection was displayed as a one – to – one correspondence function. We also extended the concept of mid-tangent points to three dimensional space and found that the similarity with the stereographic projection was retained in higher dimensions. Finally, we studied the locus of mid-tangent points of a sphere to create a mapping of the sphere to a plane.