The popular mathematical science of geometry presents a huge number of theories that are used to solve problems in a more simplified way. Our article will deal with the concept of what a theorem is, what are the most famous and often used in practice, their principle of operation, as well as give examples of when to use them.
Pythagoras’ Theorem
Rectangular triangles with sides equal to 3,4,5 were known in ancient Egypt; it was often used by people engaged in stringing. Pythagoras used algebraic methods to find the Pythagorean triples, an ordered set of three natural values. This information was generally accepted, and already Plato had given his hypothesis of finding where the Pythagorean triples combine algebra and geometry. Henceforth, the resulting theory received the name Pythagoras’ theorem, in honor of its creator.
According to the theorem, in a right-angled triangle, the length of the hypotenuse (the side lying against the angle) is equal to the sum of the lengths of the square cathetuses (the sides that form the angle). The formula is as follows:
a2+b2=c2
More than 400 proofs of the theorem were later presented, denoting its fundamental meaning. The most common rhyme used in school folklore is “Pythagoras’ pants are equal in all directions,” a name given by the comic opera Ivanov Paul.
Pythagoras’ theorem is often used to solve problems in geometry, algebra, and physics. In life, it is often used to calculate building and architectural structures.
Thales theorem
Thales’ theorem is that a pair of secant lines always forms equal segments to a pair of straight lines. This conclusion was reached by the Greek mathematician Thales of Miletus, who, according to legend, calculated the height of the pyramid of Cheops by measuring the shadow on the ground and its length. The formulation of Thales’ theorem is as follows:
(A1A2)/(B1B2)=(A2A3)/(B2B3), etc.
The Argentinean musical group “Les Luthiers” even dedicated a song to this theorem. Today, when designing various objects or models, design engineers often turn to this theorem for help.
The Sinus Theorem
The first mention of the sine theorem was in a chapter of the Almagest, but not a direct statement. Of the first ancient proofs that have come down to us on the plane, Nasir ad-Din At-Tust’s book “Treatise on the Total Quadrilateral”, written in the 13th century, is believed to be the first.
The sides of triangles are directly proportional to the sines of the opposing angles, in practice it looks like this:
(a/sin a)=(b/sin B)=(c/sin Y)
The trigonometric theory of sines is still used to this day, and is used by auto mechanics, factory workers, and even girls who draw eyebrows with a pencil.
Menelaus’ Theorem
Menelaus’s Theorem or Quadrilateral Theorem was proved in the third book of Spherika by the ancient Greek mathematician Menelaus of Alexandria. The original proof was presented for the flat case, and it was not until some time later that Menelaus transferred it to the sphere. Most theorems in project geometry are based on Menelaus’ theorem, which is formulated like this: if the points A1, B1 and C1 lie on the sides BC, CA and AB of triangle ABC then they are collinear. There are a huge number of variations of the theory, where it takes on a form depending on the direction of use:
-trigonometric equivalent;
-spherical geometry;
-Lobachevsky geometry.
The use of Menelaus’ theorem will simplify the solution of many problems and calculate the areas of figures for estimators.
The Viette Theorem
Thanks to the Vieth theorem, the coefficients of a polynomial and its roots are connected. The formulas are great for checking the correctness of finding the roots of a polynomial, as well as for composing a polynomial according to the given roots. This theorem was discovered by the French scholar François Viet while in the royal service as an advisor. The formulation of his theory is as follows:
If C1, C2, C3 are roots of a polynomial, then xn=a1xn-1+a2xn-2+an
In mathematics, Viet’s theorem is often used to solve quadratic or cubic equations using the system method. In life, when calculating apartment buildings, only specialists use it not on their own, but through special programs that perform the necessary calculations.
Many people, studying geometry at school and at universities, believe that these are rules nobody needs, although in fact they are found in various fields. Without this fundamental knowledge it would be difficult for many professionals to work, so you should not neglect the knowledge and carefully study geometry and its most common theories.