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

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.


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.

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]. 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', and ∆BCF', 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 = √3c, Then, by using cosinus directionon ∆ BXZ, ∆ CYZ, and ∆AXY as follow.
Based on cosinus directionon ∆ ABC that has been elimminated as follow Based on sinus directionon ∆ABC as follow Then, by distributing the equality (4) and the equality (6) to the equality (3) as follow If it is distributed the equality (7) to the equality (2) as follow = ( + cos ∠B) + √ ab sin∠C.
Then by distributing the equality (7) to the equality (1) as follow 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.

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.

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.

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 bx 2 + ay 2 = a 2 b 2 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°, 270 o and 360° origin quadrilateral is a rhombus. Whose value for a 45°, 135°, 225°, and 315 o 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.

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.