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This page is a help for construction of a regular 257-sided regular polygon. Pages for deriving of values of cos(k*pi/257) (pi=3.141592...) show that we can obtain these values by solving many quadratic equations.

These pages are Formulas for cos(k*pi/257). Part I, Part II, Part III, Part IV, Part V, Part VI, Part VII.

and we'll use the following property of right triangles:

Suppose AB is a diameter of the circumscribed circle on ABC. Hence, angle C is the right angle. Let CD be the height of ABC of the vertex C. Then

In other words: height of the right triangle is a geometrical mean value of intervals it divides hypotenuse.

If AB=p, h=q, then roots of our equation are AD and DB, which follows from the previously mentioned equalities of Vieta.

Exactly, we proceed the following way:

1. Take 2 intervals: one of length p, second of length q.

2. Take an interval AB of length p and circle S such that AB is a diameter of S.

3. Draw a line L parallel to the line AB at the distance q from line AB.

4. Take point D - one of points of intersection of L with S. If there are not such points - it means that equation has no roots.

5. Find point D on the line AB such that CD is perpendicular to AB.

6. AD and DB are roots of our equation.

If roots of this equation exist, they are negative. Of course we cannot construct intervals with negative lengths. That is why we construct intervals of lengths of absolute value of roots and we add few words, that roots are opposite numbers to these lengths.

To construct intervals of lengths of absolute value of roots we change the sign corresponding to coefficient p receiving an equation of Case 1. Then we proceed according to procedures of Case 1.

In this case roots have the different signs. We proceed similar way as before, now regarding the length of AB as unknown and DB equal p.

1. Build a right triangle CDE such that CD has length q, DE has length p/2 and an angle D is the right angle (90 degrees).

2. Draw line DB.

3. Draw a circle S with the center at E and radius EC.

4. Take points A and B, where S intersects line DB.

5. Length of AD is an absolute value of one root and DB is an absolute value of the second root.

6. Sign of a root corresponding to longer interval is plus, sign of a root corresponding to shorter interval is minus.

Knowing properties of right triangle, it is easy to check that equalities of Vieta are satisfied with such defined roots.

We construct roots as in the case III and then reverse signs of these roots.

where r and s are positive numbers. We want to construct intervals with ruler and compass, which have lengths equal absolute value of roots. It has only a sense if we can construct intervals of length equal r and s. Of course then we need to know a unit interval i.e. interval of length 1. Of course if we have a unit interval and for example r and s are rational, we can construct the roots.

We just reduce this case to one of previous cases plugging p=r and finding q such that

It is easy to construct such an interval q knowing mentioned above property of the right triangle. It seems to Literka that the following picture explains everything, since q is a geometrical mean value of s and 1

The hypotenuse of above right triangle consists of 2 intervals: a unit interval and interval of length s.

See the list and descriptions of mathematical pages from Mathematical Countryside.

See other pages of 'Literka' from Mathematical Countryside:

Monotonic subsequences.

Weierstrass Approximation Theorem. Bernstein's Polynomials.

A remarkable monotonic property of the gamma function .

Weight centers of simple geometrical figures.

An elementary problem can be unsolvable.

Roots of cubic equation. Cardano's formula.

Rudin's Theorem of Complex Analysis.

Exact values of trigonometric functions of angles (n*pi)/17.

Equalities for values of trigonometric functions of angles (n*pi)/17.

Factorization of a polynomial, which defines values of sine function (angles n*pi/17).

Polynomials with roots cos(2*k*pi/n).

Factorization of polynomials with roots cos(2*k*pi/n), where n is Fermat number.

Values of trigonometric functions of angles (n*pi)/257. Part I.

Values of trigonometric functions of angles (n*pi)/257. Part II, Part III, Part IV, Part V, Part VI, Part VII, Part VIII, Part IX, Part X, Part XI, Part XII, Part XIII, Part XIV.

Values of trigonometric functions of angles (n*pi)/65537. Part I.

Values of trigonometric functions of angles (n*pi)/65537. Part II, Part III and Part IV and Part V, Part VI, Part VII

Construction of a regular pentagon.

Construction of a regular heptadecagon.

Construction of a regular polygon with 257 sides.

Software that you really need.

Return to the list of pages of 'Literka' about polytopes.

Return to the main geometrical page of 'Literka'.

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