INSCRIBED CIRCLE OF GENERAL SEMI-REGULAR POLYGON AND SOME OF ITS FEATURES

Transcription

1 INTERNATIONAL JOURNAL OF GEOMETRY Vol. 2 (2013), No. 1, 5-22 INSCRIBED CIRCLE OF GENERAL SEMI-REGULAR POLYGON AND SOME OF ITS FEATURES NENAD U. STOJANOVIĆ Abstract. If above each side of a regular polygo with sides, we costruct a isosceles polygo with k-1 equal sides we get a equilateral polygo with N = (k 1) equal sides ad di eret iterior agles-semi regular polygos. Some metrical features ad relatios relatig to the iscribed circle of the geeral semi-regular equilateral polygo with N = (k 1) sides, ad with ; k 3,k; 2 N are dealt with i this paper. Furthermore, the paper cotais a proof to the theorem o geometrical costructio of the semiregular polygo with N = (k 1) sides, give a radius of a iscribed circle. 1. Itroductio Give the set of poits A j 2 E 2,j = 1; 2; : : : ; i Euclidia plae E 2, such that ay three successive poits do ot lie o a lie p ad for which we have a rule: if A j 2 p ad A j+1 2 p for each j poit A j+2 does ot belog to the lie p. 1. Polygo P or closed polygoal lie is the uio alog A 1 A 2 ; A 2 A 3 ; : : : ; A A +1 ; ad write short [ (1) P = A j A j+1 ; ( mod ) j+1 Poits A j are vertices, ad lies A j A j+1 are sides of polygo P. 2. The agles o the iside of a polygo formed by each pair of adjacet sides are agles of the polygo. Keywords ad phrases: semi-regular polygos, polygos, iscribed circle radius (2010)Mathematics Subject Classi catio: 51M04, 51M25, 51M30 Received: I revised form: Accepted:

2 6 Nead U.Stojaović 3. If o pair of polygo s sides, apart from the vertex, has o commo poits, that is,if A j A j+1 \A j+l A j+l+1 = ;; l 6= 1 polygo is simple, otherwise it is complex. This paper deals with simple polygos oly. 4. Simple polygos ca be covex ad o-covex. Polygo is covex if it all lies o the same side of ay of the lies A j A j+1, otherwise it is o-covex. Polygo P divides plae E 2 ito two disjoit subsets,u ad V. Subset U is called iterior, ad subset V is exterior area of the polygo. Uio of polygo P ad its iterior area U makes polygoal area S, which is: (2) S = P [ U 5. Give polygo P with vertices A j ; j = 1; 2; : : : ; ; ( mod ) lies of which A j A i are called polygoal diagoals if idices are ot cosecutive atural umbers, that is, j 6= i. We ca draw 3 diagoals from each vertex of the polygo with umber of vertices. 6. Exterior agle of the polygo P with vertex A j is the agle \A v;j with oe side A j+1 A j, ad vertex A j, ad the other oe is extesio of the side A j A j 1 through vertex A j : 7. Sum of all exterior agles of the give polygo P is equal to multiplied umber or product of tracig aroud the polygos i a certai directio ad 2, that is, the rule is (3) X (\A v;j ) = 2k; k 2 Z j=1 I which k is umber of turig aroud the polygo i certai directio. 8. The iterior agle of the polygo with vertex A j is the agle \A u;j ; j = 1; 2; : : : : ; for which \A u;j + \A v;j =. That is the agle with oe side A j 1 A j, ad the other side A j A j+1. Sum of all iterior agles of the polygo is de ed by equatio (4) X \A u;j = ( 2k); 2 N; k 2 Z: j=1 I which k is umber of turig aroud the polygo i certai directio. 9. A regular polygo is a polygo that is equiagular (all agles are equal i measure) ad equilateral (all sides have the same legth). Regular polygo with sides of b legth is marked as P. b The formula for iterior agles of the regular polygo P b ( 2) with sides is =. A o-covex regular polygo is a regular star polygo.for more about polygos i [4,5,6]. 10. Polygo that is either equiagular or equilateral is called semi-regular polygo. Equilateral polygo with di eret agles withi those sides are called equilateral semi-regular polygos, whereas polygos that are equiagular ad with sides di eret i legth are called equiagular semi regular polygos. For more about polygos i [1,2,3].

3 Iscribed circle of geeral semi-regular polygo 7 FIGURE 1. Covex semi-regular polygo P N with N = (k costructed above the regular polygo P b 1) sides 11. If we costruct a polygo P k with (k 1) sides, k 3,k 2 N with vertices B i ; i = 1; 2; : : : ; k over each side of the covex polygo P, 3, 2 N with vertices A j,j = 1; 2; : : : ;, ( + 1) 1 mod, that is A j = B 1 ; A j+1 = B k, we get ew polygo with N = (k 1) sides, (Figure 1) marked as P N. Here are the most importat elemets ad terms related to costructed polygos: (1) Polygo P k with vertices B 1 B 2 : : : B k 1 B k, B 1 = A j ; B k = A j+1 costructed over each side A j A j+1 ; j = 1; 2; : : : ; of polygo P with which it has oe side i commo is called edge polygo for polygo P. (2) A j B 2 ; B 2 B 3 ; : : : ; B k 1 A j+1 ; j = 1; 2; : : : ; are the sides polygo P k. (3) A j B 2 A j B 3 ; : : : ; A j B k 1 are diagoals d i ; i = 1; 2; : : : ; k 2, of the polygo P a k draw from the top A j ad that implies d k 2 = A j A j+1 = b: (4) Agles \B u;i are iterior agles of vertices B u;i of the polygo P N ad are deoted as i. Iterior agle \A u;j of the polygo of the vertices A j are deoted as j. (5) Polygo P k of the side a costructed over the side b of the polygo P is isosceles, with (k 1) equal sides, is deoted as P a k. (6) = \(d i ; d i+1 ); i = 1; 2; : : : ; k 2 deotes the agle betwee its two cosecutive diagoals draw from the vertices A j ; j = 1; 2; : : : ; for which it is true (5) = \(a; d 1 ) = \(d i d i+1 ); i = 1; 2; : : : ; k 3; d k 2 = b (7) If the isosceles polygo Pk a is costructed over each side of the b regular polygo P b with sides, the the costructed polygo with N = (k 1) of equal sides is called equilateral semi-regular polygo which is deoted as PN a.

4 8 Nead U.Stojaović 12. We aalyzed here some metric characteristics of the geeral equilateral semi-regular polygos, if side a is give, ad agle is = \(d i ; d i+1 ); i = 1; 2; : : : ; (k 2), i betwee the cosecutive diagoals of the polygo Pk a draw from the vertex Pk a of the regular polygo P. b Such semi-regular polygo with N = (k 1) sides of a legth ad agle de ed i (5) we deote as P a; N. 13. Regular polygo P b polygo is called correspodig regular polygo of the semi-regular polygo P a; N. 14. Iterior agles of the semi-regular equilateral polygo is divided ito two groups - agles at vertices B i ; i = 2; : : : ; k 1 we deote as, - agles at vertices A j ; j = 1; 2; : : : ; we deote as. 15. K N stads for the sum of the iterior agles of the semi-regular polygo P a; N. 16. S A j stads for the sum of diagoals comprised by agle ad draw from the vertex A j, ad with " A j we deote the agle betwee the diagoals draw from vertex A j comprised by agle. 17. We deote the radius of the iscribed circle of the semi-regular polygo P a; N, with r N. 2. MAIN RESULT Let o each side of the regular polygo P, b be costructed polygo Pk a, with (k 1) equal sides, ad let d l = A j B i ; l = 1; 2; : : : ; k 2; d k 2 = A j A j+1 = b; j = 1; 2; : : : ; ; i = 3; 4; : : : ; k; B k = A j+1 diagoals draw from the vertex A j,a j A j+1 = b to the vertices B i of the polygo Pk a. The followig lemma is valid for iterior agles at vertices B i ; B k = A j+1 of triagle 4A j B i 1 B i determied by diagoals d i. For more about i [7]. Lemma 2.1. Ratio of values of iterior agles 4A j B i 1 B i ; i = 3; 4; : : : ; k at vertex B i of the base A j B i = d i 2 from the give agle A j B i = d i 2 is de ed by relatio \B i = (i 2). Proof. The proof is doe by iductio o i, (i 3); i 2 N. Let us check this assertio for i = 4 because for i = 3 the claim is obvious because the triagle erected o the sides of the regular polygo is isosceles ad agles at the base b are equal as agle. If i = 4 ad isosceles rectagle is costructed o side b of the regular polygo P b (Figure 2) with vertices A 1 B 2 B 3 B 4, ad B 4 = A 2 where A 1 A 2 = b side of the regular polygo. Diagoals costructed from the vertex A 1 divide polygo A 1 B 2 B 3 B 4, ito triagles 4A 1 B 2 B 3 ad 4A 1 B 3 B 4. Accordig to the de itio of the agle we have: \B 2 A 1 B 3 = \B 2 B 3 A 1 = \B 3 A 1 B 4 = Itersectio of the ceterlie of the triagle s base 4A 1 B 2 B 3 i A 1 B 4 = b is poit S 1. Sice A 1 B 2 = B 2 B 3 = a, a ad costructio of the poit S 1 leads to coclusio that A 1 S 1 B 2 B 3 is a rhombus with side a.

5 Iscribed circle of geeral semi-regular polygo 9 FIGURE 2. Rectagle A 1 B 2 B 3 B 4 Sice B 3 S 1 = B 3 B 4 = a a triagle 4B 3 S 1 B 4 is isosceles, ad its iterior agle at vertex S 1 is exterior agle of the triagle 4A 1 B 3 S 1, thus \S 1 = 2, as well as \B 4 = 2. Let us presume that the claim is valid for a arbitrary iteger (p 1); (p 4), p 2 N, that is i = (p 1) iterior agle of the triagle 4A j B p 2 B p 1 at the vertex B p 1 has value \B p 1 = (p 3). Let us show ow that this ascertai is true for iteger p, that is for i = p. Also, iterior agle of the triagle 4A j B p 1 B p at vertex B p has value \B p = (p 2). Let us ote A j B p 2 B p 1 B p which is split ito triagles 4A j B p 2 B p 1 ad 4A j B p 1 B p by diagoal d p 3, ad that \B u;p 1 = (p 3) accordig to presumptio (Figure 3). FIGURE 3. Rectagle A j B p 2 B p 1 B p Sice iterior agles of triagles are cogruet at vertex A j, by de itio of agle, ad B p 2 B p 1 = B p 1 B p = a, it is easily prove that there is poit S such that triagle 4SB p 1 B p is isosceles triagle (Figure 3.), ad rectagle A j B p 2 B p 1 S is rectagle with perpedicular diagoals. Cogruece of triagles 4A j B p 2 B p 1 ' 4A j B p 1 S leads us to coclusio that \A j B p 1 S = (p 3):

6 10 Nead U.Stojaović Agle at vertex S is the exterior agle of triagle 4A j B p 2 B p 1. Ad thus we have \S = + (p 3) = (p 2): Sice triagle SB p 1 B p is isosceles, \B p = (p 2) which we were supposed to prove. So, for each i 2 N,i 3 iterior agle of triagle 4A j B i 1 B i at vertex B i is \B i = (i 2). FIGURE 4. Isosceles polygo Pk a costructed o side b of the regular polygo P b Lemma 2.2. Semi regular equilateral polygo P a; (k 1) with give side a ad agle de ed with (5), has iterior agles equal to a that agle ( 2) (6) = + 2(k 2) ad (k 2) iterior agles equal to a that agle (7) = 2; > 0; k 3; 3; k; 2 N Proof. Usig gure 4 ad results of lemma 2.1 it is easily prove that polygo Pk a costructed o side b of the regular polygo P b has (k 2) iterior agles with value 2, ad which are at the same time iterior agles of the semi-regular polygo PN a ; N = (k 1): So ideed, for k = 3 the costructed polygo P k is isosceles triagle with iterior agle at vertex B 2 = 2, ad for k = 4 costructed polygo is isosceles rectagle (Figure 2). That rectagle is draw by diagoal d 1 from vertex A j,j = 1; 2; : : : ;, B 1 A 1,B 4 A j+1 ad A j A j+1 = b split ito triagles A j B 2 B 3 ad A j B 3 A j+1 with iterior agles at vertices \B 2 = \B 3 = 2. Similarly it is prove that for every rectagle A j B i 2 B i 1 B i ; i = 4; 5; : : : ; k; (B 1 = A j ; B k = A j+1, A j A j+1 = b) ad the value of its vertex B i 1, \B i 1 = (i 3) + [(i 2) + ] = 2: So, i every isosceles polygo Pk a there k 2 iterior agles with measure 2 (Figure 4.). Sice isosceles polygo Pk a, is costructed o each side of regular polygo P, b it follows that equilateral semi-regular polygo PN a has total of (k 2) agles, which we were supposed to prove.

7 Iscribed circle of geeral semi-regular polygo 11 Whe iterior agle of the semi-regular equilateral polygo at vertex A j ; j = 1; 2; : : : ; is equal to sum of iterior agle of the regular polygo P b ad double value of the iterior agle of the polygo P a k at vertex B k, (Lemma 2.1) is valid \A u;j = = ( 2) + 2(k 2) which we were supposed to prove. Coditio of covexity of the semi-regular equilateral polygo P a; N the values of its agle is give i the theorem. ad Theorem 2.1. Equilateral semi-regular polygo P a; N,N = (k 1) is covex if the followig is true for the agle (8) 2 0; (k 2) k; 2 N; ; k 3: Proof. Let us write values of the iterior agles of the semi-regular polygo de ed by relatios(6),(7) i the form of liear fuctios P a; N (9) f() = ( 2) + 2(k 2); g() = 2; _ k; 2 N; _ k; 3: Sice the polygo is covex if all its iterior agles are smaller tha, to prove the theorem it is eough to show that for 8 2 0; (k 2) all iterior agles of the semi-regular polygo P a; N are smaller tha. Ideed, from this relatio = g() = 2 follows that = 0 for = 2, (Figure 5). O the basis of this ad demads > 0 ad > 0, we d that 2 (0; ) ad 0 < < 2, ad thus we have 2 0; ; k 3: (k 2) It is similar for iterior agles equal to agle, (Figure 5). If we multiply the iequality 0 < < ( 2) (k 2) with 2(k 2), ad the add to the left ad right side, we get the iequality ( 2) ( 2) ( 2) < + 2(k 2) < 2 ( 2) + ( 2) < < ; ) 2 ;, for 2 0; (k 2) ; k 3:

8 12 Nead U.Stojaović FIGURE 5. Semi-regular polygo ad covexity So, for every 2 0; (k 2) iterior agles of the semi-regular polygo P a; a; N are smaller tha. That is, semi-regular equilateral polygo PN is covex for 2 0; (k 2) : Values of the iterior agles of the covex semi-regular equilateral polygo P a; deped o the iterior agle of the correspodig regular polygo = N ( 2) true: as well as the agle. Which meas that the followig theorem is Corrolary 2.1. Covex semi-regular equilateral polygo P a; N is regular for = (k 1) ; k; 2 N; ; k 3; > 0 ad the values of its iterior agles are give i the relatio (10) = = (k 2) : (k 1) Proof. Accordig to the de itio of the regular polygo, its every agle has to be equal, thus from = ad the relatio (6),(7) we have the equatio ( 2) + 2(k 2) = 2 out of which we d out that the sought value of the agle is = (k 1) for which the semi-regular equilateral polygo P a; N is regular. O this basis we d that the value of the iterior agles is = = (k 2) : (k 1) The text further cotiues with the presetatio of some of the results regardig the iscribed circle of the semi-regular polygo ad the geometrical costructio of a covex semi-regular polygo with a give radius of the iscribed circle.

9 Iscribed circle of geeral semi-regular polygo 13 Theorem 2.2. Out of all covex equilateral semi-regular polygos with P a; N ; N = (k 1) sides costructed above the regular polygo P b with sides, a circle may be iscribed oly if k = 3,8 3; 2 N: Proof. The proof is give through two stages. Firstly, let us prove that a circle may be iscribed for P a; (k 1) for k = 3 while it may ot be possible for P a; (k 1) with k > 3, 3; 2 N, to have a iscribed circle. 1. Let us presume that a semi-regular polygo P a; (k 1) if k = 3, 3; 2 N has a iscribed circle C(O; r) (Figure.6). Let us prove that each side of the semi-regular polygo P a; 2, optioal sides a ad agle = \(a; b) to which it is covex, ad b side of the regular polygo above which it is costructed are all taget to such a circle. Let A 1 B 1 A 2 B 2 : : : A B be ( 2) vertices, ad \A i = = +2; i = 1; 2; : : : ; iterior agles to vertices A i, ad \B i = = 2,i = 1; 2; : : : ; iterior agles to vertices B i of semi-regular polygo P a; 2. Let us mark the ceter of iscribed circle C(O; r) with the mark O. If each vertex of the semi-regular polygo P a; 2 is joied with the ceter of the iscribed circle O there ca be observed the followig triagles: 4A 1 OB 1 ; 4B 1 OA 2 ; : : : ; 4A OB ; 4B OA 1 for which the followig is applicable: a) A 1 B 1 = A 2 B 2 = = A B = B A 1 = a side of the semi-regular polygo, b) \OA 1 B 1 = \OA 2 B 2 = = \OA B = 2 c) \OB 1 A 2 = \OB 2 A 3 = = \OB A 1 = 2. Thus, we may coclude that they are mutually cogruet. Let us observe oe of those triagles, e.g. 4A 1 OB 1. Let us mark its height from the poit O to side a = A 1 B 1 with h 1 while observig the said height to be equal to the radius of the iscribed circle h 1 = r. FIGURE 6. The Radius of the Iscribed Circle - Apothem

10 14 Nead U.Stojaović The cogruecy of the said triagles implies the cogruecy i their surfaces. If we take that P i = a h _ i 2 is surface of triagles 4A iob i ; i = 1; 2; : : : ; that is, of triagle 4B i OA i+1,i = 1; 2; : : : ; i mod with h i beig the height from vertex O to side a, the, from the equality i their surfaces, ad after shorteig the equatio it follows that (11) h 1 = h 2 = = h = r From this equatio we may coclude that circle C(O; r) is taget to each side of the semi-regular polygo P a; 2 i.e. it is iscribed to that semi-regular polygo (Figure 6). 2. Give that for k > 3; k 2 N; 3; 2 N for semi-regular polygo P a; (k 1) costructed above regular polygo P b with sides, there is circle C(O; r) iscribed with its ceter at poit O ad with radius r. By the de itio of the costructio of a semi-regular polygo there is a isosceles polygo Pk a of side a costructed above each side b of regular polygo P. b If the vertices of the semi-regular polygo side a become joied with poit O the polygo shall become divided ito two classes of mutually cogruet triagles i referece to the iterior agles alog the base side equal to side a. To oe class of triagles belog the followig: 4OA i B j 1 ; 4OBj k 1 A i+1; i; j = 1; 2; : : : ; ; mod which have their iterior agle alog vertex \A i = 2, ad iterior agle with vertex \B j = 2, ad base A ib j equal to side a of the semi-regular polygo (Figure 6). To the other class of triagles belog the isosceles triagles 4OBpB j j p+1, p = 1; 2; : : : ; k 3, which have their agles alog B p B p+1 = a equal to 2. From the cogruece of the rst class triagles follows the equality i their heights h a m; m = 1; 2; : : : ; 2 to base a, i.e. the followig is applicable: h a 1 = h a 2 = = h a 2 = r 1 with r 1 beig the iscribed circle radius. Similarly, from the cogruece of the secod class triagles follows the equality i their heights Ht a ; t = 1; 2; : : : ; (k 3) to base a, i.e. the followig is applicable: (12) H a 1 = H a 2 = = H a (k 3) = r 2 with r 2 beig the iscribed circle radius. Sice pursuat to presumptio r 1 = r 2 = r µwhat would oly be possible if the rst class triagles were cogruet to the secod class triagles, ad i that case this would be applicable: 2 = a; 2 ) =, i.e. meaig that polygo P(k 1) is regular, what i tur is opposite to the presumptio that it is semi-regular. Therefore, circle C(O; r) is ot a iscribed oe to the semi-regular polygo sice it is ot taget to each side of the polygo, i.e. it is taget either to base a of the rst class triagle or base a of the secod class triagle. Based o the preseted proof it follows that there may ot be iscribed a circle to a semi-regular equilateral polygo P a; (k 1) with k > 3, 3; 2 N.

11 Iscribed circle of geeral semi-regular polygo 15 Theorem 2.3. Radius of the iscribed circle of the equilateral semi-regular polygo P a; (k 1) k; 3; 2 N; which does ot have three cosecutive vertices each with its correspodig iterior agle equal to agle 2 is determied through a relatio (13) r (k 1) = a cos cos( (k 2)) si( (k 3)) Proof. Let there be vertex A 1 betwee the verticals costructed from ceter O of the iscribed circle to two eighbourig sides of the semi-regular polygo P a; (k 1), ad let there be a iterior agle as de ed i a relatio (6) correspodig to this vertex A 1, ad let there be eighbourig vertices B 1 ad B 2 with their correspodig iterior agles = 2. Now, let us observe the equiagular triagles 4OB 1 M; 4OA 1 M with OM = r (Figure 7). FIGURE 7. Radius of the Iscribed Circle It is obvious from the equiagular triagle 4OA 1 M that tg 2 = r a y, givig r = (a y)tg 2. Similarly, from equiagular triagle 4OB 1 M tg 2 = r y, givig y = r. If we replace this ad isert it ito the rst equatio we get the followig: tg 2 r tg 2 + r tg 2 = a ad out of which we d the followig: r 1 tg 2! + 1 tg = a, r = 2 a cot 2 + cot 2

12 16 Nead U.Stojaović Sice cot 2 = tg( (k 2)) ad cot 2 = tg from the previous equatio, after the processig ad shorteig of the equatio we get sought equatio: r (k 1) = a cos cos( (k 2)) si( (k 3)) Š which eeded to be prove i the rst place. Corrolary 2.2. The radius of the iscribed circle of the semi-regular polygo is give i relatio. P a; 2 (14) r 2 = a cos cos( ) si Proof. From the relatio (13) for k = 3 we get the sought relatio. Theorem 2.4. There is o covex equilateral semi-regular polygo P a; N with three cosecutive vertices each with their correspodig iterior agles equal to agle = 2, which may have a circle iscribed. Proof. Let us presume quite the opposite to this, i.e., that there is a semiregular polygo P a; (k 1) with the iscribed circle C(O; r) ad that there is a vertex B j+1 with its correspodig iterior agle = 2, betwee the verticals costructed from the ceter O of the iscribed circle oto the two eighborig sides. Let its eighborig vertices B j ad B j+2 correspod the iterior agles (Figure 7.). The, from the equiagular triagles 4OB j K, 4OKB j+1 we d that the other relatio for the radius of the iscribed circle would be as follows: (15) r (k 1) = a 2 cot Sice the equatios (13) ad (15) represet the radius of iscribed circle C(O; r), with their processig, equalig ad shorteig with a, we get the followig equatio: cos cos( (k 2)) si( (k 3)) = 1 2 cot With a presumptio that si 6= 0 ) 6= m; m 2 Z ad si( (k 3)) 6= 0, (k 3)6= l; l 2 Z ) (1 l) 6= ; k > 3; ; k 2 N (k 3) as well as with the coditio of covexity of the semi-regular polygo (Theorem 2.1), this equatio is the trasformed ito the form 2 si cos cos( (k 2)) = si( (k 3))cos, cos = 0 _ 2 si cos( (k 2)) = 0: From the equatio cos = 0 we d that the = (2l+1) 2, l 2 Z is the solutio. This solutio does ot meet the coditio of covexity of the semi-regular polygo, ot eve for oe whole umber l 2 Z.

13 Iscribed circle of geeral semi-regular polygo 17 From the secod equatio, beig that 2 si cos( (k 2)) = si( (k 1)) + si( (k 3)) we have the followig si( (k 1)) = 0: Here we d that 6= (1 l) (k 1), l 2 Z, ; k 3, ; k 2 N. The oly value of the agle which meets the coditio of covexity is the oe with l = 0, ad the = a; (k 1). Sice with such value of the agle polygo P(k 1) is regular, it follows that there is o semi-regular polygo which has three cosecutive vertices with their correspodig iterior agles equal to the agle 2 such that it ca be iscribed a circle. FIGURE 8. A circle may be iscribed to the equilateral semi-regular dodecago if k = 3, = 6. Example 1. Examples of semi-regular polygo with iscribed circles; a) For k = 3; = 6 equilateral semi-regular dodecago which may be iscribed a circle. (Figure 8). b) For k = 5; = 3 (Figure 9) ad for k = 4; = 4 (Figure 10),equilateral semi-regular dodecago which may ot be iscribed a circle. As a example, the agle value of = 10 has bee chose.

14 18 Nead U.Stojaović FIGURE 9. A circle may ot be iscribed to the equilateral dodecago with k = 5; = 3 FIGURE 10. A circle may ot be iscribed to the equilateral dodecago with k = 4, = 4.

15 Iscribed circle of geeral semi-regular polygo 19 Theorem 2.5. The ratio of surface P 2 of the semi-regular polygo P a; 2 ad the product of multiplicatio of its side a ad the radius of iscribed circle r is equal to the umber of sides of the regular polygo above which it has bee costructed, i.e. P 2 (16) ar = Proof. From the equatio for surface of the semi-regular polygo P2 a P 2 = a2 cos cos( ) si ad the formula for the radius of the iscribed circle r = r 2 = a cos cos( ) si we have the followig: P 2 = a a cos cos( ) si! = ar ) P 2 ar = with beig the umber of sides to the regular polygo above which a semi-regular polygo P2 a has bee costructed. Propositio 2.1. If the iscribed circle of semi-regular polygo P a 2 P is a uit circle, the the ratio of the umerical value of the polygo surface ad polygo side is equal to the umber of sides of the correspodig regular polygo above which it has bee costructed. P Proof. It follows from (16), if r=1, i.e. 2 a =. The followig theorem deals with the equilateral semi-regular polygos with the iscribed uit circle. Theorem 2.6. There is o semi-regular equilateral polygo P a; 2 with the iscribed uit circle ad the side a ad the whole umber legth, i.e. such that a 2 Z Proof. We have show (Corollary 2.2) that the radius of the iscribed circle is give through a relatio r 2 = a cos cos( ) si out of which, for r = 1, after the relatio processig ad shorteig, we d that the legth of a side of a semi-regular polygo P a; 2 is determied with si (17) a = cos cos( ) : If we use that what has bee show for the covex semi-regular equilateral polygos which may be iscribed a circle, that it is applicable that k = 3; 3; 2 N ad 2 (0; m ) ad that si 2 Q, = 2 ; m 2 Z or = 6 (6l1); l 2 Z, as well as that for such values of agle, si 2 (0; 1; 1 2 ) we have the followig: si, = m 2 ; m 2 Z _ = (6l 1); l 2 Z 6

16 20 Nead U.Stojaović From the rst equatio we get that = 2 m, wherefrom 2 N, oly for m = 1 ad m = 2. These values do ot meet the coditio that 3. From the secod equatio = 6 6 (6l 1) we d that = 6l1 ad that it represets a atural umber = 6 oly for l = 0. For that value there is a si 6 = 1 2. Should we replace that value i (17) we obtai the followig: 1 a = 2cos cos( 6 ) : Sice 2 cos cos( 6 ) = cos(2 6 ) + cos( 6 ) the followig sequece of iequatio is applicable: p p 3 3 < cos(2 6 ) + 2 < 2 + p 3 ) p p 1 3 < p cos(2 6 ) < 3, p p < a < 3 3 ) 4 2 p p 3 3 < a < 3 the legth of side a is ot a whole umber, what is exactly that what eeded to be prove. A geometrical costructio of the semi-regular polygo which may be iscribed a circle is give withi the followig theorem. Theorem 2.7. For a give radius r of the iscribed circle ad agle as de ed i (5) there is a equilateral semi-regular polygo P r; N, with N = (k 1) sides for k = 3; 3, 2 N which which is de ed with those elemets ad which may be geometrically costructed. Proof. Let us presume that the costructio of a such semi-regular polygo P r; 2 is possible, ad that it is preseted i Figure 11. Let C(O; r) be the iscribed circle with its ceter at poit O ad with radius OK = r. We have already show (Theorem 2.2) that out of all semi-regular polygos P a; (k 1) a circle may be iscribed oly if k = 3. Let A 1B 1 A 2 B 2 : : : A B be the vertices of a semi-regular polygo costructed above the sides of a regular polygo with vertices A 1 A 2 : : : A, ad let eighborig vertices A i i B i,i = 1; 2; : : : : have their correspodig iterior agles immediately to the vertices i the followig sequece: to vertices A i correspod agles ( 2) = + 2, ad to vertices B i correspod agles = 2. Let us take radomly two cosecutive vertices of the semi-regular polygo. For the right-agled triagles the followig is applicable: 1: For 4OKA 1 it is: \A 1 = ( 2) 2 = 2 +, \O =,\K = 2,OK = r 2: For triagle 4OKB 1 it is: OK = r,\k = 2,\B 1 = 2 ad \O = (Figure 11). Based o the give elemets r ad we ca costruct the right-agled triagle 4OKB 1. The itersectio of straight lie p through poits B 1 ; K with the agle side \O = (which may be costructed depedig o the

17 Iscribed circle of geeral semi-regular polygo 21 umber of sides of the appropriate regular polygo) determies vertex A 1 of right-agled triagle 4OKA 1. It is with this that we have costructed the side a = B 1 A 1 of the semi-regular polygo. If we itersect a taget t A1 from vertex A 1 costructed oto circle C(O; r) with circle C(A 1 ; a) we get vertex B 2. If i that vertex we costruct a taget oto a iscribed circle C(O; r) the itersectio of such taget ad circle C(B 2 ) determies vertex C(B 2 ). If we proceed further o i the same maer, we may get all the other vertices of the semi-regular equilateral polygo P r; 2. FIGURE 11. Costructio of equilateral dodecago with a give radius of iscribed circle r ad agle Costructio descriptio: Let there be give radius r ad agle. 1. We costruct a right-agled triagle 4OKB 1 with the followig give elemets: OK = r; \K = 2 ; \O = 2. We costruct a circle C(O; r),ok = r as a iscribed circle of a semiregular polygo P r; We costruct a agle i ceter O,\O =, ad the costruct a right-agled triagle 4OKA 1, the costructio of which determies side a = B 1 A 1 of the semi-regular polygo. 4.We costruct a taget from vertex A 1 to circle C(O; r) ad the we costruct vertex B 2, i the followig maer C(A 1 ; B 1 A 1 = a) \ C(O; r). 5. If we repeat the previous procedure this time from vertex B 2 we the get vertex A 2. We the proceed with the same procedure to costruct all other vertices. The example above (Figure 12) presets a costructio of the semi-regular P ;r 6 with a give radius r = 2cm of iscribed circle ad = 15.

18 22 Nead U.Stojaović FIGURE 12. Costructio P ;r 6 ; r = 2:5cm; = 15 Refereces [1] Vavilov, V.V. ad Ustiov, V.A.,Polupravylye Mogouglov a Reshetkah (Russia), Kvat, 6(2007). [2] Vavilov, V.V. ad Ustiov, V.A., Mogougolyky a Reshetkah (Russia), Moscow, [3] Hilbert, D. ad Coh-Vosse, S. Aschauliche Geometrie (Russia),Verlig vo J.Spriger, Berli, [4] Aleksadrov, A.D., Kovexy Polyedry (Russia), Moscow, [5] Radojµcić, M., Elemetara geometrija-osove i elemeti euklidske geometrije (Serbia), Belgrade, [6] Poari, P.J., Elemetaraya Geometrya (Russia), Tom 1., MCNMO, Moscow, [7] Stojaovic, N., Some metric properties of geeral semi-regular polygos, Global Joural of Advaced Research o Classical ad Moder Geometries, 1(2)(2012), UNIVERSITY OF BANJA LUKA FACULTY OF AGRICULTURE BANJA LUKA, BREPUBLIC OF SRPSKA, BOSNIA AND HERZEGOVINA sest1@gmail.com