Given its long history, there are numerous proofs (more than 350) of the Pythagorean theorem, perhaps more than any other theorem of mathematics. The proofs below are by no means exhaustive, and …

The reflexive property of congruence is often used in geometric proofs when certain congruences need to be established. For example, to prove that two triangles are congruent, 3 congruences need to be …

A conjecture is a mathematical statement that has not yet been rigorously proved. Conjectures arise when one notices a pattern that holds true for many cases. However, just because a pattern holds …

Proof by contradiction (also known as indirect proof or the method of reductio ad absurdum) is a common proof technique that is based on a very simple principle: something that leads to a …

Descartes' circle theorem (a.k.a. the kissing circle theorem) provides a quadratic equation satisfied by the radii of four mutually tangent circles. By solving this equation, one can determine the …

Finding the Incenter All triangles have an incenter, and it always lies inside the triangle. One way to find the incenter makes use of the property that the incenter is the intersection of the three angle …

Hyperbolic geometry is a type of non-Euclidean geometry that arose historically when mathematicians tried to simplify the axioms of Euclidean geometry, and instead discovered unexpectedly that …

A perfect square is an integer that can be expressed as the product of two equal integers. For example, \ (100\) is a perfect square because it is equal to \ (10\times 10\). If \ (N\) is an integer, then \ (N^2\) is …

While writing a proof by induction, there are certain fundamental terms and mathematical jargon which must be used, as well as a certain format which has to be followed. These norms can never be …

A plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analog of a point (zero dimensions), a line (one dimension), and three-dimensional space. A plane in …