Geometric Proof
In geometry, a geometric proof is a proof in which the logical steps are carried out entirely in a geometric setting. While one can prove many theorems using geometric methods, the term is usually reserved for proofs that would otherwise be very long or difficult to construct. In other words, a geometric proof is one in which the geometric details of the proof are not given for the reader to complete, but are made explicit for the reader to understand why the proof is valid. A geometric proof can be divided into three stages: The first stage of a geometric proof depends on the ability to construct figures and measure their properties. As a general rule, the properties of a figure that are important for a geometric proof are those that are invariant under the group of transformations that transform the figure in a way that respects the axioms of the system of geometry. For example, the properties of a circle are invariant under the group of affine transformations which preserve the circle's shape. The first stage of a geometric proof often involves constructing a figure using a ruler and a compass. This is usually done in two stages. First, the geometrical properties of the figure are explored, often using a sequence of proofs that are well-known in geometry. An example of a well-known proof is the proof that the sum of the angles of a triangle is 180 degrees. The second stage of constructing a figure is making an explicit construction, i.e., a drawing, or a mathematical definition, of what the figure is supposed to be. For example, a proof that the sum of the angles of a triangle is 180 degrees might proceed by having the reader construct a triangle, and then marking off the sum of the angles in degrees. In the second stage, the properties of the figure that are invariant under the group of transformations that transform the figure are examined, often using a sequence of proofs. An example of a well-known proof is the proof that the area of a circle is ? times the area of a triangle with the same base and height. In this proof, the triangle is constructed, then the area is calculated, and then the area of a circle with the same base and height is calculated (using the formula for the area of a circle). The process is repeated, this time for a circle with the same base and height. The reader is asked to check that the two areas are the same. The third stage of a geometric proof is to examine the properties of the figure that are not invariant under the group of transformations that transform the figure, and to use that knowledge to deduce the conclusion. An example of a well-known proof is the proof that the sum of the angles of a triangle is 180 degrees. The conclusion is that the sum of the angles is equal to the sum of the angles of a triangle with the same base and height. The non-invariant properties that are examined in this proof are the sum of the angles of two triangles with the same base and height.
