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From the first proof, one can see that the sum of the diagonals is equal to the perimeter of the parallelogram formed. Also, we can use vectors 1/2 the length of each side to first determine the area of the quadrilateral, and then to find areas of the four triangles divided by each side of the inner parallelogram.
The area of the parallelogram is the area of the blue region, which is the interior of the parallelogram. The base × height area formula can also be derived using the figure to the right. The area K of the parallelogram to the right (the blue area) is the total area of the rectangle less the area of the two orange triangles. The area of the ...
green area = blue area Construction for proof of parallelogram generalization. Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure shows that for a scalene triangle, the ...
dark grey area = light grey area. Pappus's area theorem describes the relationship between the areas of three parallelograms attached to three sides of an arbitrary triangle. The theorem, which can also be thought of as a generalization of the Pythagorean theorem, is named after the Greek mathematician Pappus of Alexandria (4th century AD), who ...
In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary. The result was first described by Georg Alexander Pick in 1899. [2] It was popularized in English by Hugo Steinhaus in the 1950 edition of his book Mathematical ...
Lagrange's identity for complex numbers has been obtained from a straightforward product identity. A derivation for the reals is obviously even more succinct. Since the Cauchy–Schwarz inequality is a particular case of Lagrange's identity, [4] this proof is yet another way to obtain the CS inequality. Higher order terms in the series produce ...
Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. [1] It is a theorem commonly employed in various math competitions, secondary school mathematics examinations, and has wide applicability to ...
Constructive proof. In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular ...