The conic shape, when dissected, looks like a bowl that has four flat points on one side and two flat points on the other. Any point inside the bowl can be taken to be a point inside the sphere of the conic, which is what an equivalent is. As an example, if the point inside the equivalent is your feet, it can be considered a point inside the circle formed by the four straight points, which is the sphere of the conic. If, however, you are standing in front of the equivalent, it would be a point inside your body. Because of the way this shape is dissected, it is easy to visualize it as a cone that has a point inside it at one end, but another point inside it at the other end.
Geometric shapes like the equivalent are usually used to express something as a circle or as a quadrangle. In most cases, it is not easy to see that a point inside the conic is actually part of a polyhedron or that a point inside the quadrangle is actually part of a larger polyhedron. That is why when studying a diagram showing the relationship between the sides of a triangle or the sides of a circle, it is a good idea to first look at the equivalent and then move onto the conic.
Geometry does not have to be restricted to the shapes found inside an ellipse or inside a circle. When using the equivalent of a circle as a geometry guide, it is important to consider other points of view that might make a circle or a quadrangle more useful. Other geometrical shapes can also be used as geometry guides, like polygons, hexagons, octagons and icosahedra.
Geometry is also useful when talking about surfaces such as a cube, sphere or torus. Although the equivalent of a cube is not a perfect circle or a perfect quadrangle, it does represent one point inside a sphere. The equivalent of a sphere can be compared to a rectangle or conic. So, if you were to think of the sphere as just a rectangular shape, then the equivocal of a cube would look like a hexagon.
Equidistant triangles are geometrical shapes made up of four equal sides and a common equal angle. An equidistant equiangular triangle, as its name suggests, is an equivalent of a triangle with four equal sides. These are the simplest triangles that can be constructed with only four points.
Triangle, like all triangles, can be used as geometry guides. An equivalent of a triangle can be considered as a plane when all four sides are parallel to each other. For example, if two points are placed along the equivalent, then a right angle can be drawn between them.
Equivial triangles are often used in teaching geometry. A geometric equivocal triangle may be used to represent the equator of the earth, the equator of the solar system or the equator of the universe. A triangle used as a geometry guide may be drawn in many different ways.
