Corollary

Meanings of Corollary

From the Latin corollarium, according to DigoPaul, corollary is a proposition that is deduced from what has been previously demonstrated, so it does not require a particular proof. A corollary is understood to be an obvious or unavoidable conclusion that follows from certain antecedents.

For example: “The corollary of smoking three packs of cigarettes a day is a lung disease”, “The decline of the team is the corollary of several years of poor leadership management”, “The resignation of the senator after the scandal is nothing more than the corollary of the situation that broke out last Wednesday”, “The corollary could not be different: the three protesters were released for lack of merit”.

In everyday language, a corollary appears as something logical or inescapable if the preceding facts are taken into account. A soccer player argues with the technical director of his team during training. The next day, he publicly criticizes the coach. On the third day, he goes unannounced from team practice. The corollary of this situation is that the coach disaffects the player from the squad and stops taking him into account.

In the field of logic and mathematics, the corollary is the evidence of a theorem already proven, without the need to continue investing efforts in its proof. If it is stated that all the interior angles of a square are right angles (90º) and that all squares have four interior angles, a corollary of these affirmations is that the interior angles of a square add up to 360º.

From the well -known Pythagorean Theorem, which states that the sum of the squares of the legs of a right triangle return the same value as raising the hypotenuse to the square, a corollary also emerges, which varies depending on whether we are talking about even numbers or odd. To develop this corollary, it is first necessary to establish the formula of the theorem as shown in the image.

Here you can see that the two legs are represented by the variables a and b, and that c corresponds to the hypotenuse. Based on this definition, if we have an odd number x, this Pythagorean triple can be obtained through the calculations shown in the image.

The variable a is assigned the value of x ; b corresponds to x squared, minus 1, all divided by 2; a c, similar to b but adding 1 to the square instead of subtracting it. Having understood this development, it is possible to square each component and place them in the aforementioned equality.

With respect to even numbers, if we take for example a number and, the Pythagorean triple must be formed as seen in the image. In this case, a receives the value of y ; b is assigned the square of the result of y over 2, all but 1; the value of c is similar to b, but adding 1 to the previous square. With all this, we are once again in a position to define the equality that allows us to verify the Pythagorean Theorem.

The mathematician Thales of Miletus, a native of Greece and born in the sixth century BC, bequeathed two important theorems to geometry, each with its respective corollaries. The first of the theorems establishes that if a line is drawn parallel to one of the sides of a triangle, the resulting figure will be another triangle, similar to the first one . The corollary of it is the deduction that the proportion of the sides of the new triangle is also equivalent to that of the original ones.

The second of Thales’s theorems explains that if we choose any point on a circumference of diameter AC, different from A and C, then the three will form a right triangle. Two corollaries follow from this:

1) since the distance between the center of the circumference and any of the three points of the triangle is the same, then the median of the hypotenuse (segment between the center and point B) will always measure half of the hypotenuse;

2) similar to the first, the radius of the circumference is equal to half of the hypotenuse, and the circumcenter is always located at its midpoint.

Corollary

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