Utilize the Pythagorean Theorem to formulate the equation of a circle with a specified center and radius, while the center and radius from a given equation can be found using the method of completing the square. Conceive the equation of a parabola with the assistance of a provided focus and directrix.

Formulate the equations for ellipses and hyperbolas based on the given foci by leveraging the principle of constant sum or difference of distances from the foci. Utilize coordinates for algebra-based proofs of simple geometric theorems, such as verifying whether four points in the coordinate plane form a rectangle or whether a specific point is part of a circle with a known center and point.

Establish the slope criteria for parallel and perpendicular lines, and employ them to resolve geometric challenges, such as the derivation of a line equation that is parallel or perpendicular to a known line and passes through a specified point. Discover the point on a directed line segment between two known points that divides the segment in a specific ratio.

Use coordinates to calculate perimeters of polygons and areas of triangles and rectangles, with the help of the distance formula.