Some Metric Properties and a Constructive Task of a Semi-Regular 2n-Sides Polygon

A simple polygon that either has equal all sides or all interior angles is called a semi-regular polygon. In terms of this definition, we can distinguish between two types of semi-regular polygons: equilateral polygons (that have equal all sides and different interior angles) and equiangular polygons (that have equal interior angles and different sides). To analyze the metric properties of semi-regular polygons, knowing only one basic element, e.g. the length of a side, as in regular polygons, is not enough. Therefore, in addition to the side of a semi-regular polygon, we use another characteristic element of it to analyze the metric features, and that is the angle δ=∠(a,b) between the side of a semi-regular polygon PN and the side b of its inscribed regular polygon Pn. Some metric properties of a semi-regular equilateral 2n-sides polygon are analyzed in this paper with respect to these two characteristic elements. Some of the problems discussed in the paper are: convexity, calculation of surface area, dependence on the length of sides a and δ, calculation of the radius of the inscribed circle depending on the sides a and angles δ, and calculation of the surface area in which the radius of the inscribed circle is known, as well as the relationship between them. It has been shown that the formula for calculating the surface area of regular polygons results from the formula for the surface area of 2n-side semi-regular, equilateral polygons. Further, by using these results, it has been shown that the cross-sections of regular polygons inscribed to semi-regular equilateral polygons, the vertices of equiangular semi-regular polygons, as well as the sides of the regular polygons inscribed to it, intersect in the same manner at the vertices of the equilateral semi-regular polygon. It has further been shown that the sides of the equiangular semi-regular polygon refer to each other as the sines of the angles created by the sides of the inscribed polygons and the side of the semi-regular polygon.


Introduction
A simple polygon . . . that has equal all sides or equal all interior angles is called a semi-regular polygon [8,9,10,11]. In terms of definition, we distinguish between two types of semi-regular polygons: equiangular (having equal interior angles and different sides) and equilateral (having equal sides, and different interior angles). We consider that vertex , 1, 2, . . . , , of a polygon is in an even position, or odd position, if index is even, or odd number, respectively. In this paper we consider convex equilateral semi-regular polygons. The marking is as follows: 1. is a number of sides in a semi-regular equilateral polygon, 2. is a number of sides in a regular polygon, 3. is a side in a semi-regular polygon , 4. , side in a regular polygon "inscribed" to a semi-regular polygon . . . , constructed by joining its vertices in even (or odd) positions, 5. Interior angles of a semi-regular polygon at odd vertices are marked with , and those at even vertices are marked with 6. Isosceles triangles , , … , are triangles constructed over each side of an "inscribed" regular -triangle. 7. ! is the radius of the inscribed circle to the semi-regular equilateral polygon (Figure 1).
In addition to these interpreted marks, all other marks will be interpreted when mentioned in a given definition. To a semi-regular equilateral polygon . . . with = 2 ⋅ , ≥ 2, ∈ ℕ and with equal sides there can be "inscribed" regular − % &' polygons by joining odd vertices ≡ . . . , or even vertices ≡ . . . . To analyze the metric properties of regular polygons, it is sufficient for us to know one basic element, i.e. the length of a side, while for the semi-regular polygons this is not sufficient [2].
Therefore, in addition to side of a semi-regular polygon, for the analysis of the metric properties we will use another element of it, and that is the angle between side of the semi-regular polygon and side of its "inscribed" regular polygon, which we mark with *, i.e. * = ∠( , ) (Figure 1) [3].
To show that a semi-regular equilateral 2 −sides polygon is given by side and angle *, we write: (,) .
, ≥ 2 is the interior angle of the "inscribed" regular polygon , then = -+ 2* = ( ). + 2* gives interior angles at odd vertices, and = 0 − 2* gives the ones at even vertices of the semi-regular polygon of a side , where * = ∠( , ) marks the angle between the sides of polygons and ( Figure 1). Here, we consider that a regular polygon with = 2 sides (segment) is "inscribed" to a semi-regular equilateral quadrilateral (rhombus).
Next, we consider those values of angle * for which is a convex semi-regular equilateral polygon. We find the values of angle * for which semi-regular equilateral polygon , ≥ 2, ∈ ℕ is convex from the inequality which connects the definition of convexity with the values of interior angles of the semi-regular polygon [2]. That is, from inequality < 0 i < 0 and values of angles , we find that for all * ∈ 〈0, . 〉 a semi-regular equilateral polygon is convex, while for * = . it is convex and regular. The following is true: An equilateral semi-regular 2 -sides polygon of a side and angle * is: 1. convex, if * ∈ 〈0, . 〉. For * = . it is regular and convex.

〉.
3. if * > . , then semi-regular polygon is not defined [8][9][10][11]. Let us show that the surface area of ( , *) equilateral semi-regular 2 -sides polygon as a function of a side and angle * is calculated by the formula (1). Theorem 1. The surface area of equilateral semi-regular 2n-side polygon of side and angle * is calculated by the of formula

Surface of Semi-Regular Equilateral
where is the number of sides of the "inscribed" regular polygon ≥ 2, ∈ ℕ. Proof: Note that for the surface area of the inscribed equilateral semi-regular polygon (,) the following equality is valid where D(E F ) is the surface area of the inscribed regularsides polygon of a side , and D(E ) is the surface area of the isosceles triangle. Let us further note a fragment of a semi-regular polygon ( Figure 2). It follows from a special right triangle △ G that = 2 cos*, so the surface area of an isosceles triangle is D(E ) = % *cos*. Further, for the surface area of the "inscribed" polygon , according to the markings in Figure 2, the following is valid D(E F ) = 4 cot 2 = 4 (2 cos*) cot 0 = cos *cot 0 .
Based on that, the surface area of a semi-regular equilateral polygon is  The latter equality is a formula for calculating the surface area of a regular 2 -sides polygon.
If in the formula for calculating the surface area of a semiregular polygon we include that 2, then we get that i.e. we get a formula for calculating the surface area of an equilateral quadrilateral (rhombus), where * is the angle between the diagonal and the side of the rhombus.

Radius of the Inscribed Circle
For a semi-regular equilateral convex 2 -sides polygon of a side and angle * to which the circle of radius ! can be inscribed, the following theorem holds: Theorem 3. The radius of a circle inscribed to a semiregular equilateral polygon (,) depending on a side and angle * is calculated by the following formula where is the number of sides of the "inscribed" regular polygon and # 2 and . If we include the calculated value for V in the latter equality we find that ! + $ ! tan 2 , tan+ 0 2 $ *,.
If we express radius ! from this equality, after calculation, we find that From this equality, we obtain the required form  ? @ and substitute it in the formula for the surface area of a semiregular equilateral 2 $sides polygon, after calcuation we get the following: From this it follows that the surface area of a semi-regular equilateral polygon is equal to the product of the length of the side, the radius of the inscribed circle, and the number of sides of the inscribed regular -sides polygon. , after calculation we find that the radius of the inscribed circle to a semi-regular equilateral polygon depending on a side of a regular polygon inscribed to it and angle * ∠+ , ,, can be expressed by the following relation where is the side of the inscribed regular -sides polygon, ! is the radius of the inscribed circle and * ∠+ , , is the angle between the side of the semi-regular polygon and the inscribed semi-regular -sides polygon.

A Constructive Task
Let us formulate and prove the following theorem, which is related to the problem of a semi-regular 2 -sides polygon and regular polygons inscribed therein. , > 2, ∈ ℕ because for = 2 we have a rhombus to which it is not possible to inscribe a semi-regular equiangular polygon. From equiangular triangle △ c Q we find that the side of the inscribed regular is given in a relation and that the other side of equiangular 2n-sides polygon ℬ is given in a relation , > 2, ∈ ℕ Based on relations (13) and (15), we find that and select the value of angle * such that it is possible to construct it geometrically, e.g. in the form of * = . @ . Note that from the Bernoulli's inequality (1 + V) ≥ 1 + V, (which for V = 1, transforms into a form of 2 ≥ 1 + > ), it follows that inequalities In the set of natural numbers, only solution = 4 meets the requirement > 2.

Conclusion
Based on the content presented in this paper, it follows that the basic metric properties, such as: the convexity, the surface area of a semi-regular 2n-sides polygon, the radius of the inscribed circle and the ratios of a semi-regular 2n-sides polygon can be expressed by side a and angle δ.