International Journal of Theoretical and Applied Mathematics
Volume 3, Issue 2, April 2017, Pages: 54-57

Development of Napoleon’s Theorem on the Rectangles in Case of Inside Direction

Mashadi1, *, Chitra Valentika2, Sri Gemawati1

1Analysis and Geometry Group Department of Mathematics, Fakulty of Mathematics and Natural Sciences University of Riau, Bina Widya Campus, Pekanbaru, Indonesia

2Fakulty of Mathematics and Natural Sciences, University of Riau Bina Widya Campus, Pekanbaru, Indonesia

(C. Valentika)
(S. Gemawati)

*Corresponding author

Mashadi, Chitra Valentika, Sri Gemawati. Development of Napoleon’s Theorem on the Rectangles in Case of Inside Direction. International Journal of Theoretical and Applied Mathematics. Vol. 3, No. 2, 2017, pp. 54-57. doi: 10.11648/j.ijtam.20170302.11

Received: October 25, 2016; Accepted: January 12, 2017; Published: February 9, 2017

Abstract: In this paper will be discussed Napoleon’s Theorem on rectangles that has two parallel pair sides of the square case that built inside direction. The theorem will be proven by using congruence approach. At the end of Napoleon's theorem was discussed the development of Geogebra application in case of inside direction.

Keywords: Napoleon’s Triangle, Napoleon’s Theorem on Quadrilateral, Inside Directions and Congruence

1. Introduction

Geogiev and Mushkarov [2, 6, 7, 10, 11 and 12] stated that Napoleon’s theorem was discovered by Napoleon Bonaparte (1769-1821), a French emperor and mathematics figure in geometry. After four years he was death, the theorem was published for the first time by W. Rutherford in case of equilateral triangle that constructed in the outside direction [1, 3, 4, 6 and 7]. Then, developed by Wetzel [5, 6 and 7] in case of an equilateral triangle that constructed in the inside direction. Napoleon's theorem in direction is on each side of any triangle constructed equilateral triangle leads to the inside or outside of the third point in the center of the equilateral triangle will form a new equilateral triangle. It is called the Napoleon triangle [3, 6 and 7]. In accordance with Napoleon's Theorem on the triangle then the theorem will be developed on a rectangle. In this article the author discusses the proof of quadrilateral Napoleon’s theorem is a quadrilateral that has two pairs of parallel sides such asaparallelogram, and the developement of applying Napoleon Theorem with Geo-Gebra application.

2. Napoleon’s Theorem on Triangle

Look at picture 1, on the AB side was constructed ABD equilateral triangle, and on the BC side was constructed BCF equilateral triangle, and on the AC side was constructed ACE equilateral triangle, the equilateral triangles were constructed to inside direction [4, 6 and 11]. For example point P, Q, and R are center point of equilateral triangle. The center points form equilateral triangle that can be called as inside Napoleon’s triangle [4, 5, 6 and 7]. The Napoleon’s theorem on the triangle in case of inside direction was provided as follow [3, 4, 7, 8 and 9].

Figure 1. Napoleon’s Theoremon traingle in case of inside direction.

Theorem 1. On this case explained equilateral trianglethat constructed on each side of ABC quadrilateral to inside. For example X, Y, and Z are center point of ABD’, ACE’, andBCF’, the points form equilateral triangle that can be called as Napoleon’s inside triangle, the illustration is showed on figure 1.

Proof: Picture 1 is the illustration of the proof of Napoleon’s theorem on the triangle that leads inside direction.

Furthermore, in providing XYZ is equilateral triangle, will be shown that XY=YZ=XZ in accordance with trigonometry. By using basic trigonometry formula as follow

BX= AX =c,

AY= CY =b,

BZ= CZ =a.

Then, by using cosinus directionon BXZ, CYZ, and AXY as follow.

cos (,

(cos sin ,

ac cos +

ac sin

(1)

cos (

(cos sin ,

abcos +

absin,

absin.(2)

cos (.

(cos sin ,

bccos +

bc sin,

bcsin. (3)

Based on cosinus directionon ABC that has been elimminated as follow

+ bc cos  = cos ,(4)

+ ac cos  = cos .(5)

Based on sinus directionon ABC as follow

sin , (6)

sin . (7)

Then, by distributing the equality (4) and the equality (6) to the equality (3) as follow

= bc(sin)

=  absin(8)

If it is distributed the equality (7) to the equality (2) as follow

=  ab sin. (9)

Then by distributing the equality (7) to the equality (1) as follow

=  ac (sin),

=  absin (10)

Based on the equality of (8), (9) and (10) it is clear that XY = YZ = XZ, so it can be inffered that XYZ is equilateral triangle.

3. Napoleon’s Theorem on the Quadrilateral

Napoleon’s theorem on the quadrilateral is discussed on the quadrilateral that has two pairs of parallel sides, one of them on the parallelogram. Look at the picture 2, on the AB side is constructed ABHG square, on AD side is constructed ADEF square, on the CD side is constructed CDKL square, and on the BC sided is constructed BCIJ square. Then, each of square is constructed into inside direction. Furhtermore, each of square’s center point is connected then can be called as Napoleon’s inside quadrilateral.

Theorem 2. It is provided quadrilateral that form as parallelogram ABCD. On each of parallelogramside is constructed ADEF, ABGH, CDKL, and BDKLsquares that lead into inside. For example M', N', O', and P' are each of square centre point that constructed into inside direction. If they are connected they will form M'N'O'P' square.

Proof. To showing M'N'O'Pis square, it can be proved that M'N' = N'O'= O'P' = M'P', and P’M'N =M’P'O’ = 90°. Look at picture 2, GD line and BF are deducted, for example deduction point is S and T.

Then for the example W point is deduction point BC and AF line. Look at picture 2, TBS = BFW, FBW = BTS, then the three angles are equal FWB = TSB. Because of ADEFsquare is rotated 180°, so the ADEF square is straight with BCIJ so it is caused FWB = TSB = 90°. pull the line N’P’ and M’O’ so it cuts in a point, for example R point. Because FN’//BP’then it cause TSB = GVR =P’RO’ = 90o. look at M’AN’ and O'DN’, M’A=O’D, M'AN’ = O'DN’, N’A = N’D. then we obtained that M’AN’ and O'DN’ is congruence. So that M'N = N'O’. it cause M'N’O’ isosceles triangle so M'N’R = O'N’R=45°. It is clear that P’M'N =M’P'O’ = 90°. So, it is proved that quadrilateral  is square.

Figure 2. The Proof of Napoleon’s Theorem on the Quadrilateral.

4. Development of Napoleon’s Theorem on the Quadrilateral

Development of the quadrilateral Napoleon’s theorem developed based on quadrilateral parallelogram to a square in case of leads into inside direction.

Theorem 3. Given a quadrilateral parallelogram ABCD, and on each side leading into the square built. Then draw a line FG, EL, KJ, and HI. For example point Q, R, S, and T is the midpoint of the fourth line. If the four points are connected, the square formed QRST.

Proof. For example point Q, R, S, and T is the midpoint of the line FG, EL, KJ, and HI. To show QRST is square it will be proved TQ = QR, and TQR = 90°. Figure 3, draw a line from point Q to point S and point R to the point Q. So the lines QS and RQ lines intersect at one point, said point U. Before Show TQ = QR will at first show UT = UR. Figure 3, UY = UT, YT = VR so UT = UR. Then Note ΔQUT and ΔQUR, UT = UR, TUQ = TUQ and UQ = UQ thus obtained TQ = QR. From Theorem 2, VU = UY, and VUY = 90°, then also obtained QTR = 90°, so it proved QRST quadrilateral is a square.

Figure 3. Development of Napoleon's theorem on the quadrilateral.

5. Developmen of Napoleoan’s Theorem with Applications Geo-Gebra

Napoleon's Theorem development is performed by using Geogebra application. Georgiev and Mushkarov [12] stated that application Geogebra is dynamic mathematics software that can be used as a tool in the learning of mathematics. To apply Theorem Napoleon with applications GeoGebra namely by inputting equations elliptical bx2 + ay2 = a2b2 on Graphics 1, then make four points on the graph the ellipse is to enter A = (a cos α, b sin α), B = (a cos (α + 90°), bsin (α+90°)), C = (a cos (α + 180°), b sin (α + 180°)), D = (a cos (α + 270°), b sin (α + 270°)) on Graphics 1. for a whose value 0°, 90°, 180°, 270o and 360° origin quadrilateral is a rhombus. Whose value for a 45°, 135°, 225°, and 315o origin quadrilateral is a square or rectangular depending chart major and minor axes of the ellipse. As for the other angles of a quadrilateral origin formed is parallelogram.

Furthermore, to make the square leads into, select the Regular Polygon can be seen from the way of construction, hover the cursor back to the image and select quadrilateral. To make a point of the center of each square choose Midpoint or center can be seen from the way mengkontruksinya, hover the cursor on the second point of the square diagonal [2, h.7]. Then connect the four points of the square center, forming a quadrilateral in. Note ilutrasi Geogebra application in Figure 4.

Figure 4. Napoleon’s Theorem with Geogebra application.

6. Conclusion

After several experiments Napoleon’S theorem in the quadrilateral thus obtained Napoleon's theorem applies only to the quadrilateral which possess two pairs of parallel sides like a square, rhombus, rectangle, parallelogram. Napoleon's theorem on the line leading inside the case is if the square was built on each side, the fourth point square center will be forming a square called the Napoleon quadrilateral. Proof of that is done by using the concept of congruence. Development of Napoleon on a quadrilateral theorem can be developed to form a square on the midpoint of the line so as to form a new square.

References

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