Napoleon s theorem states that if we construct equilateral triangles on the sides of any triangle all outward or all inward, the centers of those equilateral triangles themselves form an equilateral triangle, as illustrated below. Triangle abc has equilateral triangles drawn on its edges. Napoleon s theorem on each side of a given arbitrary triangle describe an equilateral triangle exterior to the given one, and join the centers of the three thus obtained equilateral triangles. We present some analogous theorems and related statements for the isotropic galilean plane. Oct 07, 2008 no matter what shape the green triangle has, the red triangle is always equilateral. Let us explore when the original triangle is an isosceles triangle. An interactive column for maa online that uses a java applet to simulate a puzzle or mathematical problem, one not stated. The classical napoleons theorem states that the centers of the equilateral triangles which were built to the sides of any triangle are the vertices of an equilateral triangle. Napoleon s theorem states that if one takes any triangle, such as the solid one on the left in the figure below, and constructs equilateral triangles on each side whose lengths are the same as the length of each side of the original triangle shown on the right as dotted triangles, the centers of the equilateral triangles will themselves form. Napoleons theorem napoleons theorem states that if you take a general triangle and draw an equilateral triangle on each side, then the triangle formed by joining the incenters of these new triangles is equilateral. Napoleons theorem states that if one takes any triangle, such as the solid one on the left in the figure below, and constructs equilateral triangles on each side whose lengths are the same as the length of each side of the original triangle shown on the right as dotted triangles, the centers of the equilateral triangles will themselves form. In geometry, napoleons theorem states that if equilateral triangles are constructed on the sides. Napoleon s theorem is the introductory page to a dozen other pages with various observations and generalizations. Show that the resulting triangle is also equilateral.
The wolfram demonstrations project contains thousands of free interactive visualizations, with new entries added daily. Pdf the aim of this work is to use napoleons theorem in different regular polygons, and decide whether we can prove napoleons theorem is. On napoleons theorem in the isotropic planeon napoleons. Points p, q and r are the centres of the equilateral triangles.
This article incorporates material from theorem on planetmath, which is licensed under the gfdl. Anecdote about napoleon bonaparte 17691821 napoleon was known as an amateur mathematician. There is just a single phrase in parentheses without any named source. Some theorems on polygons with oneline spectral proofs. A generalization of the napoleons theorem awesomemath. In many recent publications the result is called napoleon s theorem. Napoleons theorem states the fact that if equilateral triangles are drawn outside of any triangle, the centers of the equilateral triangles will form an equilateral triangle. Free modern geometry books download ebooks online textbooks. For more information, films, and interactive material, see. Napoleons theorem states that the centers of the three outer napoleon triangles form another equilateral triangle. The triangles ba 1c, cb 1a, ac 1bare constructed possibly degenerate on the sides of the triangle abcsuch that all of the three triangles are either externally or internally and fellowing the conditions. Napoleons theorem on quadrilateral developed based napoleons the orem on quadrilateral for case square built leads outward.
On napoleons theorem in the isotropic planeon napoleon. Permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. Napoleons theorem, on each side of a given arbitrary triangle describe an equilateral triangle exterior to the given one, and join the centers of the three thus obtained equilateral triangles. Converses of napoleons theorem article pdf available in the american mathematical monthly 994. Napoleon s theorem states that the centers of the three outer napoleon triangles form another equilateral triangle.
Napoleon was known to be an amateur mathematician, but it is not known if he either created or solved the problem. Pdf convolution filters for triangles semantic scholar. The following facts are relevant to the assignation of napoleon s name to the result. Napoleons original theorem refers to arbitrary triangles in the euclidean plane. Miranda napoleonlike theorems transformation geometry we would like to prove napoleons theorem by.
Media in category napoleons theorem the following 10 files are in this category, out of 10 total. The classical theorem of ceva, ceva, menelaus and selftransversality, the general transversality theorem, the theorems of hoehn and prattkasapi, circular products of ratios involving circles, circle transversality theorems, a basic lemma and some applications, affinely regular polygons, linear transformations. Pdf napoleons theorem in regular polygons researchgate. Napoleons theorem by kala fischbein and tammy brooks. Construct external equilateral triangles on the sides of a triangle. On the sides of a triangle construct equilateral triangles outer or inner napoleon triangles. Rutherford, a famous english mathematician, never mentioned napoleon in his initial printing and, in fact, napoleons name didnt even appear in a. In these equilateral triangles, we can then find the centers.
The first edition of the novel was published in 1937, and was written by napoleon hill. Maccools proof of napoleo ns theorem a sequel to the maccoolwest point 1 m. Pdf think and grow rich book by napoleon hill free. Maccools proof of napoleons theorem a sequel to the.
In many recent publications the result is called napoleons theorem. In the second chapter of his book davis gives free rein to his imagination and presents an amusing, tongueincheek explanation for the naming of napoleon s theorem. Napoleons theorem, doing as much as you can in your head. Napoleon s theorem is an extremely detailed and complete treatment of napoleon s theorem, including a delightful tessilation of the plane that results from the construction of a simple proof of the theorem. Rutherford, a famous english mathematician, never mentioned napoleon in his initial printing and, in fact, napoleons name didnt even appear in a publication relating to the theorem until 1911. Some insights on the topic related to convex quadrilaterals will be made. On converses of napoleons theorem and a modified shape. Media in category napoleon s theorem the following 10 files are in this category, out of 10 total. A generalization of the napoleons theorem khakimboy egamberganov abstract in this article we present a generalization of the napoleons theorem. The centers of the three equilateral triangles erected on the sides of the arbitrarily given original triangle, all outward or all inward, are the vertices of the new equilateral triangle. In this exceptional case there is a free choice for the third vertex. The construction of a new triangle by erecting similar ears on the sides of a given triangle as in napoleons theorem can be considered as the convolution of the initial triangle with another triangle. It is rather doubtful that the emperor actually discovered this theorem, but it is true that he was interested in mathematics.
A median of a triangle is any line from a vertex to the midpoint of the opposite side. A copy of the license is included in the section entitled gnu free documentation license. No matter what shape the green triangle has, the red triangle is always equilateral. Or alternatively, consult my book available in printed form or pdf download at some. Given any triangle, we can construct equilateral triangles on the sides of each leg.
The challenge is to divide the circle into four equal arcs using only a compass. Since considering the kolob theorem my mind has not rested. Each of these centers is in the same location because the triangles are equilateral. The main characters of this self help, personal development story are. However, its earliest attribution to him comes from an italian textbook of elementary geometry by faifofer published in 1911, almost a century after napoleons death. Licensing edit permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. It is known that napoleon bonaparte was a bit of a mathematician with a great interest in geometry. If equilateral triangles are externally erected on the sides of a given triangle, then their three corresponding circumcenters form an equilateral triangle. Isosceles triangle on a segment let the points a and b have af. Considered to be one of napoleon s greatest victories 66,000 french troops battled 85,000 russoaustrian troops napoleon intentionally exposed a weak right flank to draw the enemy attack over there, sucking troops out of the centre once the enemy was committed, he had reinforcements.
We are interested in the following neat result, referred to as napoleons theorem named after emperor napoleon bonaparte, also well known as an amateur mathematician. This is said to be one of the mostoften rediscovered results in mathematics. It is known that napoleon bonaparte was a bit of a. Napoleons theorem has been given many different proofs. The distance from this midpoint to c is equal to 1 2abtanwith this we. On the optimality of napoleon triangles springerlink. Napoleon buonaparte is known as military stratege and imperator, but he was also a chess player and amateur mathematician. Mathematical encounters of the second kind is a joyful memoir of the author s encounters, some actual and some fictional, with a number of mathematicians and historical figures. In geometry, napoleon s theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centres of those equilateral triangles themselves form an equilateral triangle. Napoleon s original theorem refers to arbitrary triangles in the euclidean plane. Pdf the aim of this work is to use napoleon s theorem in different regular polygons, and decide whether we can prove napoleon s theorem is only. In the second chapter of his book davis gives free rein to his imagination and. The three centroids of the triangles so constructed define an equilateral triangle. We use the discrete fourier transformation and a shape function to give a complete and explicit description of such convolution filters and their iterates.
Modern geometry gilbert lecture notes download book. Napoleons theorem and generalizations through linear maps. Napoleons theorem was initially published in the ladies diary in 1825, four years after napoleons death, by william rutherford. Napoleonlike theorems napoleons theorem transformation. Laplace even briefly served as his minister of interior for 6 weeks, but did not leave a good impression. Free download or read online think and grow rich pdf epub book. If equilateral triangles bcp, caq, abr are erected externally on the sides of. Pdf in this article we discuss napoleons theorem on the rectangles having two pairs of parallel sides for the case of outside direction. Smyth i came across this incredibly short proof in one of maccools notebooks. One proof, beautiful in its generality, is based on the theory of spiral similarities. According to wetzel 62, the result is surely one of the mostoften rediscovered results in mathematics. The wonder of it all is that the plan of salvation is set before us in the night sky. The green triangle, which is napoleon s triangle, is still an equilateral triangle.
Wetzel interesting converse results in elementary geometry can often be found by taking certain parts of a figure as given in position and investigating the extent to which various other parts of the figure are determined. The statement also holds for the three inner triangles. Maccools proof of napoleons theorem a sequel to the maccool. Synthesis and related investigations we recall a syntheticgeometric demonstration of napoleon theorem, which makes use of the fermat point. Davis has won numerous awards for his scientific writing, among them the national book award. Napoleon i, emperor of the french, king of italy, mediator of the swiss confederation and protector of the confederation of the rhine, born napoleon bonaparte 15 august 1769 5 may 1821, was a general during the french revolution, the ruler of france as first consul premier consul of the french republic from 11 november 1799 to 18 may 1804, emperor of the french empereur des francais. The following facts are relevant to the assignation of napoleons name to the result. If you get stuck, here are some things to meditate upon. There is a historical anecdote about napoleon who was emperor of the french. Then three free vertices of these triangles form a triangle similar to the original two.
On each side of a triangle, erect an equilateral triangle, lying exterior to the original triangle. Most proofs of it are understandable by a good high school student. Memoirs of napoleon 2 will, with the other additions and improvements already alluded to, tend to give it a place in every well. Napoleons friend the italian mathematician lorenzo mascheroni introduced the limitation of using only a. What led coxeter and greitzer to disparage napoleons abilities isnt clear, though it may have been just a poor effort at humor. Napoleons theorem in this handout we show how complex numbers could be used to solve pure plane geometry questions. Napoleon s theorem, on each side of a given arbitrary triangle describe an equilateral triangle exterior to the given one, and join the centers of the three thus obtained equilateral triangles.
Introduction the fourier decomposition of a planar or nonplanar 4 polygon and circulant matrices have been used for a long time in the study of polygon transformations. Several writers expressed some degree of disbelief in the justiability of this attribution, but most did not dare to acknowledge explicitly that there is no basis whatsoever for this eponymy. Napoleons theorem napoleons theorem states that if we construct equilateral triangles externally on the sides of any arbitrary triangle, the centroids of those equilateral triangles also form an equilateral triangle. An interactive column using java applets napoleons thm and the napoleon points weisstein, eric w. Some use simple euclidean geometry so, for example, 4, p. Holmes 31 states the result in 1874 as a fact without refer.
Then the segments connecting the centroids of the three equilateral triangles themselves form an equilateral triangle. Napoleons theorem requires logical thinking but little more. This is a theorem attributed by legend to napoleon bonaparte. Mathematical encounters of the second kind is a joyful memoir of the authors encounters, some actual and some fictional, with a. An elementary geometric construction, known as napoleons theorem, produces an equilateral triangle, obtained from equilateral triangles erected on the sides of any initial triangle. Proving napoleons theorem department of mathematics. Napoleons problem is a compass construction problem. Jul 10, 2009 the wolfram demonstrations project contains thousands of free interactive visualizations, with new entries added daily. He established such institutions as the ecole polytechnique with a view to training military engineers, but these institutions benefited. This theorem is generally attributed to napoleon bonaparte 17691821, although it has also been traced back to 1825 schmidt 1990, wentzel 1992, eddy and. Now, let s see what happens when our original triangle is a right triangle. Take any triangle, and for each edge construct an equilateral triangle with that edge as its base.
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