Which method of finding exoplanets uses the planet and stars gravitational "tug of war" in order to detect he planet itself?

Gravity

Explanation:

Gravity is the force that keeps planets in orbit around the Sun. ... A planet's size and mass determines its gravitational pull. A planet's mass and size determines how strong its gravitational pull is.

from almost the creation of the first true maps of the earth, people started seeing how continents would be able to fit together.   in particular, people noticed that south america fits almost exactly into africa.

you should be aware that while world maps were around early in the 1600’s and better defined by the end of the 1700’s, those maps were not ‘public’ but were treated as state secrets.   and so it was not till much later that the broader ‘science’ community had access to good quality and accurate world maps.

get the whole story here…

it is now known that most of the major continental masses can be fit together in a jigsaw process.   in fact we now know that the continents were indeed once all joined together as one land mass – the supercontinent of pangaea.

if you look at most world maps you will ponder how this is evidence, as the continents really don’t appear to ‘fit’ together very well.   what you need to understand is that the vast majority of maps are drawn using a map projection that has the north and south poles (which are points) are a line the width of the map at the top and bottom.   this means that the map distorts the true shape of the earth’s land and oceans as you love closer to the poles.     some other map projections distort less, but have the map shape looking like segments of an orange.   see here for a range of map projections.   check out the way greenland changes size in each!

what is more remarkable, is the next piece of evidence when added to the shape of continents.

location of mountains and fossils

what is remarkable, when you reassemble all the continents together, the ancient mountain ranges (many now eroded to their cores) and the location of rock types and fossils all match up.