No projection of the sphere preserves straightness and area

Question

I have been studying map projections (i.e. a homeomorphic embedding of a neighbourhood of a sphere into a plane or cylinder). Lambert projection preserves area, stereographic and Mercator projection preserve angles, and gnomic projection preserves straightness (in the sense that arcs of great circles are sent to straight line segments). I am interested in whether there is a projection which has two of these properties. My nose tells me this is known not to be the case. I was easily able to show that no projection may simultaneously preserve straightness and angles, but got stuck for the other two cases. I am hoping someone can suggest some good literature on this matter.

Answer 1

Building off Jephph's comment (wherein they note that there is no map projection preserving area and angles or angles and straightness), there is no map projection preserving area and straightness.

One way to see this is to take Lexell's Theorem (I can't seem to find a good reference site to link, but a quick search will find it). If we have a fixed base $\overline{AB}$ of a triangle on the sphere, and some third point $C$, the set of points $X$ where the area of $\triangle ABX$ is equal to the area of $\triangle ABC$ is a circle passing through $C$ and the antipodal points of $A$ and $B$.

However, in Euclidean geometry, the set of such points is a straight line parallel to the base $\overline{AB}$. Thus, any map projection that preserves area must send some circles (i.e. the Lexell circle of points $X$) on the sphere onto lines in the plane. As such, the projection cannot preserve straightness.