Mercator preserves local angles
For the spherical Mercator projection, the local length multiplier is sec(latitude), and the area multiplier is its square. At 60° latitude, lengths are enlarged by two and areas by four. A shortest great-circle route usually bends on this map.
Equal area is a different promise
Cylindrical equal-area maps balance east–west and north–south stretching so their product stays one. Gall–Peters uses standard parallels at ±45°. Preserving area does not preserve every shape, bearing or distance.
Equidistant from one centre
A north-polar azimuthal-equidistant map preserves distance along radii from the North Pole. Its tangential scale grows with angular distance from that centre. Two southern cities can therefore have a severely distorted connecting distance even though pole-to-city radii are correct.