You could create a custom Chamberlin trimetric projected CRS.
From Snyder, John P. Map projections: A working manual. USGS Publications Warehouse. 1987:
The Chamberlin Trimetric projection is an approximate "three-point equidistant" projection, constructed so that distances from three chosen points to any
other point on the map are approximately correct. The latter distances cannot be
exactly true, but the projection is a compromise which the National Geographic
Society uses as a standard projection for maps of most continents. This projection
was geometrically constructed by the Society, of which Wellman Chamberlin
(1908-76) was chief cartographer for many years.
For the Wikimedia example:
Africa on Chamberlin trimetric projection. 10° graticule, anchor points at (22°N, 0°), (22°N, 45°E), and (22°S, 22°30'E).
You can create the custom PROJ.4 string for the Chamberlin trimetric conversion method for a WGS84 datum:
+proj=chamb +lat_1=22 +lon_1=0 +lat_2=22 +lon_2=45 +lat_3=-22 +lon_3=22.5 +datum=WGS84 +type=crs
And ask its WKT2:2019 string with:
C:\>projinfo "+proj=chamb +lat_1=22 +lon_1=0 +lat_2=22 +lon_2=45 +lat_3=-22 +lon_3=22.5 +datum=WGS84 +type=crs"
PROJ.4 string:
+proj=chamb +lat_1=22 +lon_1=0 +lat_2=22 +lon_2=45 +lat_3=-22 +lon_3=22.5 +datum=WGS84 +type=crs
WKT2:2019 string:
PROJCRS["unknown",
BASEGEOGCRS["unknown",
DATUM["World Geodetic System 1984",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]],
ID["EPSG",6326]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8901]]],
CONVERSION["unknown",
METHOD["PROJ chamb"],
PARAMETER["lat_1",22,
ANGLEUNIT["degree",0.0174532925199433,
ID["EPSG",9122]]],
PARAMETER["lon_1",0,
ANGLEUNIT["degree",0.0174532925199433,
ID["EPSG",9122]]],
PARAMETER["lat_2",22,
ANGLEUNIT["degree",0.0174532925199433,
ID["EPSG",9122]]],
PARAMETER["lon_2",45,
ANGLEUNIT["degree",0.0174532925199433,
ID["EPSG",9122]]],
PARAMETER["lat_3",-22,
ANGLEUNIT["degree",0.0174532925199433,
ID["EPSG",9122]]],
PARAMETER["lon_3",22.5,
ANGLEUNIT["degree",0.0174532925199433,
ID["EPSG",9122]]]],
CS[Cartesian,2],
AXIS["(E)",east,
ORDER[1],
LENGTHUNIT["metre",1,
ID["EPSG",9001]]],
AXIS["(N)",north,
ORDER[2],
LENGTHUNIT["metre",1,
ID["EPSG",9001]]]]