The area of the figure by the points of the formula. The formula for the pik at the school course in planimetry
ESSAY
BEHIND THE TOPIC “Bagatokutniks on the gates. FORMULA PIKA.»
Trained by student 9 "B"
Baranova Oleksandra
Entry
Klіtchasty papyr (more precisely - yogo vuzli), on which we often respect for better painting and chairs, є one of the most important butts of dotted grates on the flat. Even so, this simple graph served K. Gauss as the right point for leveling the area of the stake with a number of points with integer coordinates that are located in the middle. Those who made simple geometric assertions about figures on a plane may have deep traces in arithmetic calculations, were clearly marked by G. Minkovsky in 1896, if they had obtained geometric methods for viewing number-theoretic problems.
Grati on the flat with an effortful way, which allows you to shift the analytical tasks of my geometrical mine and back. Rukh on this unique bridge between analysis and geometry, becoming intense and bilateral.
Grati on the square and in the open space
Let's look at two families of parallel lines on the plane, which divide the plane into equal parallelograms; Anonymous L all points of the cross line of these straight lines (or the impersonal vertices of all parallelograms) are called point vertices or simply vertices, and the points themselves are called vertices of the vertex. Whether any of these parallelograms is called a fundamental parallelogram or a parallelogram that generates a lattice.
You can set the grati like this. Let's say that on the square
given two straight lines l0 and m0 that overlap, as well as two positive numbers a and b. Along the sides in the straight line l0, we draw parallel lines l±1, l±2, l±3, on the lines a, 2a, 3a, in the direction of it. Similarly, along the edges of the straight line m0 on the lines b, 2b, 3b, ... we draw the straight lines m±1, m±2, m±3,…
Significantly all crossing points of lines l i with lines m j; impersonal all points of crossbar and є lattice L
It is important to mothers on the vazі, scho grati are folded into points (vuzlіv), and themselves
do not lie straight up to it. One and the same Grati can be taken away
for the help of different families of parallel lines.
Significantly, a number of the simplest dominions of dotted dots.
1. A straight line, to pass through two knots of the gate, to avenge innumerably rich knots of the gate. At the same time, everything is between the sudo nodes, which lie on this straight line, equal to each other.
2. The transformation of the parallel transfer of the area (space), which translates one node of the lattice into the next node, translates the Grati in itself.
3. Expand the centrally symmetrical way in the middle of a winding, like a cross two knots of tsikh ґrat. More than that, in the middle of all the cracks and knots at the knots of the grating, they establish a new grating, which includes the old one.
4. (Rule of the parallelogram.) If the three vertices of the parallelogram are the knots of the ґrat, then the fourth vertex of the parallelogram is the same vertex of the ґrat. At the expanse: if only the tops of the parallelepiped, which do not lie in the same plane, are the knots of the grat, then the other peaks are also the knots of the grat.
5. Just like a parallelogram with vertices at the nodes of the gate, do not avenge other nodes on the sides and in the middle of itself, in order to generate a lattice, tobto. є її fundamental parallelogram. More than that, power is the criterion for the fact that the parallelogram is fundamental.
The so-called orthogonal grid Z 2 is depicted on the little one, which is formed by points with integer coordinates in the Cartesian coordinate system. The same family of points can be taken away by a reline of other families of lines, which are not orthogonal. In this way, the graph of the point without a middle is not related to the family of straight lines on the front of the fundamental parallelogram.
Correct bagatokutniki on the gates
Trikutnik that square.
Theorem 1:
The correct knitting pattern cannot be worked out on the whole solution Z 2 .
Proof:
Let's assume that if there is a correct knitting pattern, it is possible to solve it in the necessary order and that the cob of coordinates is located in one of the vertices, and the coordinates (a, b) and (c, d) can be found in the other vertex. It is possible to consider that the numbers a, b, c, d do not have double digits, double digits ±1. It remains to be seen that the points (0, 0), (a/k, b/k), (c/k, d/k) are also the vertices of a regular tricutnik, since k is a double digit of two or more numbers.
Skіlki a 2 +b 2 \u003d c 2 + d 2 \u003d (a−c) 2 + (b−d) 2, then it is possible that a 2 + b 2 \u003d c 2 + d 2 \u003d 2 (ac + bd) . Also, a 2 +b 2 +c 2 +d 2 = 4(ac+bd), so. the sum of the squares of all the numbers is divisible by 4. Altogether, all of the numbers are paired, and all of them are unpaired. The first is impossible for those who, in our choice, are mutually simple. It’s impossible for a friend that even then the spiving of a 2 + b 2 \u003d (a-c) 2 + (b-d) 2 is not victorious, because the left part is not divided by 4, but the right is divided. Otriman super-accuracy and bring the formulation of hardness.
There is more rich evidence for the correct trikutnik.
It’s clear that the correct six-piece can’t be broken
on the solution Z 2 , so in the other fall, having crossed one of the vertices through one, we would take away the correct trikutnik, stowing on the solution, which, as we know, is impossible. However, in the space on the gates Z 3 it is possible to roztashuvat like a correct three-piece, so a correct six-piece. Enough to present the correct six-piece.
Theorem 2:
There are no flat gratings to avenge a square and a regular tricot at the same time.
Proof:
Let's not accept it, tobto. that on deyakih grats L it is possible to scatter the correct tricot T=ABC and square K=APQR at the same time.
Starting from the square K, let's get the lattice L0. Oskilki tg 60 ◦ =√3 irrational number, then one by changing "en":["We3LG8pK-LU"],"es":["SOucI5iDT6A"],"pt":["miKNlJlSW5Y","D5zN1ht6Plg" ])
