Applied and Computational Mathematics 2020; 9(4): 112-117 http://www.sciencepublishinggroup.com/j/acm doi: 10.11648/j.acm.20200903.18 ISSN: 2328-5605 (Print); ISSN: 2328-5613 (Online)

Some Metric Properties of Semi-Regular Equilateral Nonagons

Nenad Stojanovic

Department of Mathematics, Faculty of Agriculture, University of Banja Luka, Banja, Luka, Bosnia and Herzegovina Email address:

To cite this article: Nenad Stojanovic. Some Metric Properties of Semi-Regular Equilateral Nonagons. Applied and Computational Mathematics. Vol. 9, No. 4, 2020, pp. 112-117. doi: 10.11648/j.acm.20200903.18

Received : May 14, 2020; Accepted : June 2, 2020; Published : June 17, 2020

Abstract: A simple polygon that either has equal all sides or all interior angles is called a semi-regular nonagon. 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). Unlike regular polygons, one characteristic element is not enough to analyze the metric properties of semi-regular polygons, and an additional one is needed. To select this additional characteristic element, note that the following regular triangles can be inscribed to a semi-regular equilateral nonagon by joining vertices: ∆A1 A4A7, A2 A5 A8, A3 A6 A9. Now have a look at triangle A1 A4A7. Let us use the mark φ= (a,b 1) to mark the angle between side a of the semi-regular nonagon and side b 1 of the inscribed regular triangle. In interpreting the metric properties of a semi-regular equilateral nonagon, in addition to its side, we also use the angle that such side creates with the side of one of the three regular triangles that can be inscribed to such semi-regular nonagon. We consider the way in which convexity, possibility of construction, surface area, and other properties depend on a side of the semi-regular nonagon and angle φ= (a,b 1). Keywords: Polygons, Semi-Regular Nonagon. Surface Area, Convexity Poligons

be an angle between side of the semi-regular 1. Introduction nonagon, and side of the inscribed regular triangle Note that g where marks. the A simple polygon that has equal either 2, , , , . . . angle between diagonal of isosecles quadrilateral all sides or all interior angles is called a semi-regular nonagon. In terms of this definition, we can distinguish drawn from vertex and side , or side . In this way, in addition to side between two types of semi-regular polygons: equilateral , , polygons (that have equal all sides and different interior of the semi-regular nonagon, we also get the second characteristic1,2, … ,9 element, angle , which we need in angles) and equiangular polygons (that have equal interior , angles and different sides), [5, 8-13]. We consider vertex order to analyze the metric properties [3-5]. In this paper, the of a semi-regular equilateral nonagon to be in following marks are used: an ,even, 1, or . . .in ,9 an odd position, if index is an even, or an odd 1. the side of a semi-regular polygon is marked with number, respectively. (Figure 1). In this paper, we consider 2. the number of sides of the “inscribed” regular triangle is the metric properties of convex equilateral semi-regular marked with nonagons. Unlike with regular polygons, the length of one of 3. the side of a regular triangle ”inscribed” to a semi- the sides is not sufficient for analyzing the metric properties regular nonagon, constructed by . . . of equilateral semi-regular polygons, and another joining its vertices in even (or odd) positions is marked characteristic element is needed, [2, 3, 7]. For the selection of with - , i=1,2,3, [1, 4]. this element, note that the following regular triangles can be 4. the interior, angles of a semi-regular nonagon at the inscribed to a semi-regular equilateral nonagon: vertices of the "inscribed" regular triangle (Figure 1). Let us look at the are marked with , and at the other vertices of inscribed, equilateral , triangle,, i.e. . Let the semi-regular, nonagon, they are marked with , 113 Nenad Stojanovic: Some Metric Properties of Semi-Regular Equilateral Nonagons

1. Isocescles quadrilaterals at vertices , the following applies are "edge" quadrilaterals constructed, above, , , , , , (2) each side of the "inscribed" regular triangle φ. , , (Figure 1), [8-13]. The relation between the angles which are created by the 2. is a diagonal drawn from vertex of an isosceles side of the inscribed regular triangle and the side of the semi- quadrilateral constructed above side regular nonagon (Figure 1) is given in a Theorem. of the "inscribed" regular triangle Theorem 1. Let (Figure 1),[1.3.4]. , , , be the regular triangles inscribed to semi-regular In addition to these marks, all other marks will be equilateral nonagon . Then the following interpreted when mentioned in a given definition. To a semi- equality applies for angles , . . . , , , regular equilateral nonagon with equal sides φ a, b ψ a, b ω . . . between the sides of the inscribed triangles and side there can be “inscribed” three regular triangles, i.e.: a, b of the semi-regular equilateral nonagon . In interpreting the , , metric properties of the semi regular nonagon in this paper, in (3) addition to side we also use angle which this side closes with side of a regular triangle that is inscribed to Proof: Let regular equilateral triangles, as marked in , it, and for which the following applies: (Figure Figure 1, be inscribed to a semi-regular equilateral nonagon. 1). To show that a semi-regular equilateral nonagon , is given by Then, note that the following condition is valid for the side and angle we write ,. interior angles of isosceles quadrilateral (and quadrilaterals ) constructed above, the , sides of inscribed regular triangle : 2. My Result (4) 2.1. Convexity of a Semi-regular Equilateral Nonagon 2 First, let us analyze the dependence of the internal angles which then, according to markings in Figure 1, takes the form of: of a semi-regular nonagon , (dependence on parameter ), and then the manner in which convexity depends on the (5) value of this parameter. For this purpose, let us use mark to 2 mark its interior angles at the vertices of regular triangle where: which is “inscribed” to it, and constructed by connecting vertices , and let us use mark to mark . the interior angles at, other, vertices of the semi-r egular equilateral nonagon (Figure 1). From equality (5) we get that Note that the following is valid for the interior angles of the semi-regular equilateral nonagon at vertices . , , (1) This is similar for the interior angles of isosceles 2 2φ, quadrilateral constructed where is the interior angle of the regular triangle, while above the sides of the , inscribed regular, triangle, and all for the interior angles equal to angle angles in inscribed regular triangle . For example, for the interior angles of a quadrilateral the following applies , (6) 2 According to markings (Figure 1), the equality (6) has the form of from which we find 2 2 that equality is valid. As for the other quadrilaterals, it also turns out that the equality is valid. In case of the interior angles of isosceles quadrilaterals it also shows in a similar procedure, that the, equality is valid, i.e. . Consequence 2. If all angles between the sides of the inscribed regular triangles and side of the semi-regular equilateral nonagon are equal, then the nonagon is regular. Figure 1. A semi-regular nonagon – marks and elements. Applied and Computational Mathematics 2020; 9(4): 112-117 114

Proof. From equality for , it regular equilateral nonagon defined by relations (1) and (2) are at intervals; and . Since follows that . For this value of angle the interior , , , angles of the semi-regular nonagon are equal, i.e. , in for all values the semi-regular , 0, other words, it is regular. equilateral nonagon is convex. If the equilateral semi- The manner in which the convexity of the semi-regular regular nonagon is convex and regular, with internal angles nonagon depends on the value of angle is shown in a Theorem. . If then, starting from inequality , , φ by elemental transformations we get that the values of the interior angles belong to intervals , and , and , respectively, and the semi-regular , nonagon is convex. For the value of the internal angles of a semi- , π regular nonagon equal to angle are at interval , and , 3 the angles equal to angle are at interval 0, Consequently, it folows that a semi-regular nonagon is not convex, given these values of angle . Let Then, for the interior angles equal to angle disappear, . and for it is in other words, a semi-regular nonagon disappears, i.e. it 0, transforms into other polygons Figure 2. Convexity of a semi-regular nonagon in relation to value of angle 2.2. Construction of a Semi-Regular Nonagon . Theorem 3. A semi-regular equilateral polygon Let us analyze the possibility of constructing a semi- , with interior angles defined by relations (1) regular equilateral nonagon if the length of its side and and (2),. side . . and angle , where is the side angle are given, where is the side of an , , , of inscribed regular triangle is: convex for all values of "inscribed" regular triangle . angle Let us suppose that the problem has been solved and that (for the nonagon is convex and regular, the semi-regular nonagon is the required one as shown in 0, Figure 3. Let be the inscribed regular triangle and that is not the subject of this paper), non-convex for all and be the side of this traingle. A quadrilateral values of angle and not defined for (it || , π π constructed above side of regular degenerates). triangle is an isosceles one. Proof: Let us present the interior angles of a semi-regular The rays of that quadrilateral are equal to the given side . equilateral nonagon as linear functions of parameter . Let An isosceles triangle is an auxiliary triangle which us say that can be constructed as it is (rule ASA),: || || and and On a straight line we φ. 2, φ. choose point and construct a given angle 3 φ , and its perpendicular bisector , and we get angle . A graph of function – let us mark it as – cuts across Γ Then ) and ) . graph of function at point ( with abscissa being Vertex of quadrilateral, is the, intersection of Γ , , (Figure 2). For this value of angle , the nonagon is circle and side of angle , and its base , p φ regular. Further, using a definition of a polygon convexity, is the side of inscribed regular triangle ( ), we get the value of angle . Given the , 0 If we . construct an equilateral triangle over side we intersection point of straight lines (Figure 2) and the value of get vertex . Further, above ray of the "inscribed" angle we can distinguish between the following: , regular triangle we construct an isosceles quadrilateral such when and that at vertices and we construct the given angle and 0, when and its bisector, so that one ray of that angle is base . The , following applies to vertices : when , , π and where and are when . , bisectors of angle constructed at vertex . If we repeat Let us consider these in order: the process by constructing an isosceles quadrilate ral over For the values of the interior angles of a semi- side , we get vertices 0, , . 115 Nenad Stojanovic: Some Metric Properties of Semi-Regular Equilateral Nonagons

Construction and its description: and angle between the sides of inscribed , We complete the geometric construction of a semi-regular regular triangle and side of the semi-regular nonagon with a given length of side and angle as nonagon is calculated in the following formula follows: , (7) 1 2cosφ cot Proof: Let semi-regular equilateral nonagon , be given (Figure 1). The following applies for … surface area ,

, 3 ,

where is the surface area of the inscribed equilateral triangle of side , and is the surface area of the isosceles quadrilateral constructed above side of the inscribed triangle, the side of which is equal to the length of side of semi-regular nonagon ,. Let be an isosceles quadrilateral (Figure 4) constructed over side .

Figure 3. Construction of a semi-regular nonagon with the given side and angle . 1. Construct the given angle at vertex , where is its ray, and half-line is its other ray. 2. Construct an auxiliary isosceles triangle on the grounds of the known elements: || , Figure 4. Isosceles polygon constructed above side . using the elementary rule of ASA. ,3. Construct φvertex as follows; Note that triangle is an isosceles one. Therefore, we get height , and side , . from a special right triangle sinφ . Since the surface 12cosφ area of 4. Construct an equilateral triangle of side parallelogram is and the surface area of Thus, we also get vertex because is the triangle . sinφ isosceles triangle is , based on that, we find that is "inscribed" to a semi-regular nonagon. sin2φ 5. Vertex is the intersection of a bisector of angle that the surface area of isosceles quadrilateral constructed at vertex and circle i.e. constructed above side is equal to , , Thesinφ surface1 cosφ area. of equilateral triangle (Figure , 1) is 6. Construct vertex as the intersection of a bisector of angle that is constructed at vertex and circle √3 i.e. 1 2cosφ . , 4 If we include the obtained values in the starting equality, , we get 7. Construct vertices and similarly as vertices and with a note that the construction is to be compelted , 3 on side, By a similar procedure, construct vertices The construction. of a semi-regular nonagon with the √3 given, length. of side and angle is shown in Figure 3. 1 2cosφ 3 sinφ1 cosφ 4

2.3. Surface Area of a Semi-Regular Nonagon The manner in which the surface area of a semi-regular 3 nonagon is related to the given side and angle is shown 1 2cosφ cot 16sin cos in the Theorem. 4 3 2 2

Theorem 4. The surface area of equilateral semi-regular nonagon , with the given length of side … Applied and Computational Mathematics 2020; 9(4): 112-117 116

that: and and 3 3sinφ sin3φ . 1 2cosφ cot . As the angles at vertices and are equal, we conclude 4 3 2sin 2 that is an isosceles triangle. Let be the point of The last equality is obtained if we notice that contact between side and the inscribed circles with radius , then and let be the contact point of side and the inscribed circle, and let . If there sin cos 42sin cos 32 2 2 2 2 is an inscribed circle, it is unique and its radii are equal i.e. 16sin cos 2 2 2 sin 2sin Let us show that equality is not true and that 2 2 Note . the special right angle triangle The. 4sin φ following is true for its elements . 2sin 2 3 √ 3sinφ 3sinφ 4sin φ 3sinφ 3sinφ 4sin φ 1 2cosφ, φ, 3 6 2sin 2sin 2 2 φ. 3 3sinφ sin3φ .◊ 2sin 2 Consequence 5. The surface area of a regular nonagon with the given side is , (8) 6.181 Proof. Since the following is valid for the regular nonagon: . If in the formula for the surface area , 3 3sinφ sin3φ 1 2cosφ cot 4 3 2sin we include the value of angle , after calculation,2 we , get , The resulting value is equal to the 6.181 value obtained when using formula to calculate the surface area of the regular Figure 5. Semi-regular nonagon and inscribed circle. nonagon6.18182 of side . Since we find that √ 2.4. Circle Inscribed to a Regular Nonagon cos φ 1 A problem related to the construction of the inscribed 2cosφ cos φ. Let us write , because triangle is not circle to a convex equilateral semi-regular nonagon is || considered in a Theorem. an isosceles one. Then . From special right || Theorem 6. A circle cannot be inscribed to semi-regular angle triangle it follows that tan φ. equilateral convex nonagon , Similarly, from special right angle triangle it follows Proof. Let us suppose, on the contrary, that a circle can be that . inscribed. Let there be given semi-regular equilateral For| isosceles| triangle it follows that nonagon , (Figure 5) and let point be the center of the and , and for special right angle triangle it inscribed circle, and let be an isosceles is and quadrilateral constructed above side of the inscribed regular triangle . The interior angles of a and the following is valid || If we equate the last equalities, we get semi-regular nonagon at vertices are equal to angle , and they are equal to angle at . 2, φ As by assumption (there is an inscribed circle), vertices and . Let us construct triangles: , from Note, that . because: we have the quadratic equality √ , and . Based on the congruence of these triangles, it follows 2 0 4 φ 117 Nenad Stojanovic: Some Metric Properties of Semi-Regular Equilateral Nonagons

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