Lets start with the basic nail - the square. Many of you are thinking, "yeah yeah, I can do a square nail", but the square nail is actually the basis in all other shapes so let's review it. To best file a square nail to shape, use a medium-grit file to shape the side walls first. File the side wall straight up and then change the angle to blend. Repeat this on the other side. Then, holding the file straight against the edge of the nail - either vertically or horizontally - file the free edge straight across. Then turn the client’s hand around so her fingers are pointing "up" and you are looking at the nail and straighten the free edge from that angle When looking at it the file should be perpendicular to the nail to achieve the hard square. One tip I was given many years ago that has stayed with me to this day is to keep your elbow by your side when filing sidewalls! If your elbow sticks out, your sidewalls will not be straight. Its simple but so effective.
Once you have a square nail, it is just a matter or changing the angle of your file at the free edge to start shaping into most other shapes (extreme shapes such as duck foot, lipstick or edge are the exception to this rule). As an example, for a squoval nail, instead of holding the file at a 90 degree angle to the nail, we angle it down a bit. The more we angle, the rounder the tip becomes and if we hold it nearly flat under the nail we end up with an oval nail.
Watch the videos below and watch how Elaine Watson angles the file to get the perfect shape to the tip.
Filing Almond Nails
Filing Square Nails
Filing Round Nails
Similarly, ballerina or coffin nails would be done the same way except in stead of coming to a point at the center, the angles would end the same distance from the center. You could visualize a longer triangle that ends off the free edge OR you could start with a stiletto and then "cut off" the tip, but you would have to start with making the nails longer to begin with.
The below video show sculpting a stiletto nail and filing into shape.
Build and File a Short Stiletto Nail
I hope this helps you start thinking about shaping the nails in a different way, By using your high school geometry you can learn to make some gorgeous shapes! (and you said you would never use geometry!)