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MAC = NBC = 900 (on construction). These corners are entered and leaning on one and a touch an arch (in our case, on a semi-circle) therefore points of O1 and O2 coincide and lie on a piece of DC (DC – ADB bisector).

Still in the ancient time practiced for different needs approached on a structure of any regular polygon: So, for example, Heron Alexandriyski finds approximate value of the party correct nine squares.

The sophism is the sequence of the statement, reasonings, constructions containing the hidden mistake at the expense of what it is possible to draw an incorrect conclusion. The task usually consists in finding a mistake in reasonings.

All arch of a circle of radius of R is divided into 4 big and 4 small parts which alternate one by one. The most part is twice longer than the small. To determine the area of an octagon which tops are points of division of an arch of a circle.

On diameter of a circle the equilateral triangle of AVS is under construction. Diameter of AV is divisible on 9 equal parts. Connecting the second point of division to triangle top With, we will continue a straight line before crossing with a circle in V. Luga point of AV is the ninth part of a circle, AV chord the party of the correct nonagon.

In the paper creative, independent work of the pupil is looked through. The paper differs in good selection of tasks which the teacher can use at lessons and in individual work with the most prepared pupils.

For example, it is possible to divide a circle into 17 equal parts and into 257 equal parts, as 17 and the 257th essence prime numbers of a look + 1(17 = + 1; 275 = +. The proof of it goes beyond elementary mathematics.

Let two nonparallel direct an and b be given. From points And and In these straight lines we will put perpendiculars before crossing in the Page point. Through three points And, In and With we will draw the circle crossing a straight line and in the M point, and b straight line in N point. On creation of MAC = NBC = 900, so these corners rely on diameters of MS and NC of the constructed circle. The middle of these diameters – points of O1 and O2 – the centers of the same circle.

The problem of creation of the correct n-square is reduced to division of a circle into n of equal parts. One practical reception of such division was offered by the French mathematician N. Bion. This reception is as follows: let it is required to divide a circle, for example, into 9 equal parts (see drawing).

That fact that the entered corner relying on diameter — a straight line, was known to Babylonians 4000 years ago. Its first proof is attributed by Pamphylia, the Roman writer of times of Neron, to Thales Miletsky. Some commentators of Euclid believe that Thales's proof based on the offer that the sum of corners of a triangle is equal 2d, there was a following: having designated corners with a diameter through 1, 2, and parts of a corner of AVS on which it is dissected by OS radius, through 3, 4, we receive, on the one hand:

The circle can be divided into any other number of equal parts approximately. Let, for example, it is required to divide a circle into 7 equal parts. Then previously we will calculate the size of the central corner, it :. We cannot construct just the same corner, but on a protractor approximately we can postpone at the center a corner in 510 and then we will receive approximately part of a circle.