Nine point circle proof pdf

Now using the euler line theorem, we are able to prove the ninepointcircle theorem which states that in any triangle, the midpoints of the sides, the feet of the. Abstractthe ninepoint circle theorem is one of the most beautiful and surprising theorems in euclidean geometry. It is so called because it passes through nine significant points of the triangle, among which the simplest to construct are the midpoints of the triangles sides. The nine point circle also known as eulers circle or feuerbachs circle of a given triangle is a circle which passes through 9 significant points. Because of these different names, there have been misunderstand among mathematicians about the ninepoint circles history. One of the mysterious features is the ninepoint circle. A generalization of the ninepoint circle and euler line. Let o be the center of the circumcircle c and let n be the center of the nine point circle d.

There are midpoints galore in this problem in fact, six of the nine points that we are interested in are defined as midpoints. Let point d be the midpoint of side ab, point e be the midpoint of side ac, and point f be the midpoint of side bc note triangle def is the. International journal of mathematical education in science and technology. The next three points are created from the midpoints of each of the triangles sides g, h, i. Jan 23, 2016 for the love of physics walter lewin may 16, 2011 duration. Nt nl nf of the circumscribed circle and 1 radius 2 ni nj ne of the circumscribed circle. In 1822 karl feuerbach proved that the ninepoint circle is tangent to the incircles and excircles of the triangle. A hinged realization of a plane tessellation java a lemma of equal areas java a lemma on the road to sawayama. The nine point circle is often also referred to as the euler or feuerbach circle. If, for instance, we consider a triangle abc with its orthocentre h as an orthic fourpoint, any proof that shows that the ninepoint circle touches the inscribed or an escribed circle of the triangle abc, will, in general, also show that it touches the inscribed and. The earliest author to whom the discovery of the nine pointcircle has been attributed is euler, but no one has ever given a reference to any passage in eulers writings where the characteristic property of this circle is either stated or implied. Trigonometrycircles and trianglesthe ninepoint circle. The first three points are the feet of the altitudes of our triangle with the name of d, e, and f.

Let abc be a triangle with orthocenter h and nine point center n. This paper describes the heuristic discovery and partial proof of a generalization of the famous nine point circle to a nine point conic, and its associated euler line. The ninepoint circle of a triangle is tangent to the incircle and each of the three excircles of the triangle. Guided discovery of the ninepoint circle theorem and its. Thus, b and b as well as c and c are inverse images with respect to our inversion transformation. In any triangle, three remarkable points circumcenter, centroid, and orthocenter are collinear, that is, lie on the same line, eulers li. The nine point circle of a triangle is tangent to the incircle and each of the three excircles of the triangle. This paper describes the heuristic discovery and partial proof of a generalization of the famous ninepoint circle to a ninepoint conic, and its associated euler line. The nine point circle of a triangle is a circle going through 9 key points. However, in trigonometry material of senior high school mathematics, the concept of triangle circumcircle is used for proving sine rule. Feuerbachs theorem, including the first published proof, appears in karl. The ninepoint circle owes its discovery to a group of famous mathematicians over the course of about 40 years, though it is most generally though perhaps not most fairly attributed to karl feuerbach, a german mathematician who rediscovered it in the nineteenth century however it was known even to euler dorrie, 100 great problems.

The ninepoint circle satisfies several important and. The ninepoint circle main concept the ninepoint circle, also known as eulers circle or the feuerbach circle, is a figure that can be constructed using specific concyclic points defined by any given triangle. Three natural homoteties of the ninepoint circle forum. The fact that the ninepoint circle exists at all is amazing in itself. The ninepoint circle is another circle defined from a triangle. Introduction how would you draw a circle inside a triangle, touching all three sides. In this note, we give a simple proof of feuerbachs theorem using straightforward vector computations.

But every triangle has three bases, and if we consider. These three triples of points make nine in all, giving the circle its name. Pdf the concept of circles is an ancient concept that has appeared since. The black lines are construction of the points, the orange lines are construction of the circle s. Since b and c are on the 9points circle, and the 9pts circle passes. If a is a right angle, where does it lie in relation to the ninepoint circle. The orthocentre h, the nine point circle centre n, the centroid g and the circumcentre o of any triangle lie on a line known as the euler line. On the ninepoint conic proceedings of the edinburgh. Let point j be the foot of the altitude from point c to side ab, point k be the foot of the altitude from point b to side ac, and point l be the foot of the altitude from point a to side. Pdf a generalization of the ninepoint circle and euler. The main purpose of the paper is to present a new proof of the two celebrated theorems. The earliest author to whom the discovery of the ninepointcircle has been attributed is euler, but no one has ever given a reference to any passage in eulers writings where the characteristic property of this circle is either stated or implied. The circle through the midpoints of the sides passes through the base points feet of the. The theorem that states that n touches the incircle internally and the excir cles externally is due to feuerbach 1822 feu.

The nine point circle passes through these three midpoints. The ninepoint circle theorem claims that the nine points lie on a circle centered at n, the midpoint of d and h, on the euler line. The fact that the nine point circle exists at all is amazing in itself. Three natural homoteties of the ninepoint circle 211 theorem 3. That circle which passes through the feet of the altitudes of a given triangle. The center of any ninepoint circle the ninepoint center lies on the corresponding triangles euler line, at the midpoint between that triangles orthocenter and circumcenter. Origin so called because the circle passes through nine points of interest. The radius of the ninepoint circle is half that of the circumcircle, and its centre bisects the line between the circumcentre and the orthocentre. A result closely associated with the ninepoint circle is that of the euler line, namely that the orthocentre h, centroid g, circumcentre o. The center of the nine point circle is the midpoint of the line segment joining the orthocenter and the circumcenter, and hence lies on. The nine point circle is a circle which appears in any triangle, made up of midpoints and intersections of perpendicular lines that cross through points.

A radius is an interval which joins the centre to a point on the circumference. Ninepointcircle dictionary definition ninepointcircle. Create the problem draw a circle, mark its centre and draw a diameter through the centre. This paper describes a set of instructional activities that can help students discover the ninepoint circle theorem through investigation in a dynamic geometry. The radius of the ninepoint cirlce is r 2, where r is the circumradius radius of the circumcircle. Nov 15, 2016 let point d be the midpoint of side ab, point e be the midpoint of side ac, and point f be the midpoint of side bc note triangle def is the medial triangle. Elementary proof of the existence of a circle through nine points of a given triangle. The proof will use the line wy as the base of the triangle.

Proof by the eight point circle theorem 2 note that quadrilaterals abch, abhc, and ahbc all have perpendicular diagonals. Let o be the center of the circumcircle c and let n be the center of the ninepoint circle d. The black lines are construction of the points, the orange lines are construction of the circles. Pdf ninepoint circle, pedal circle and cevian circle quang. Triangles with ninepoint center on the circumcircle begin with a circle, center o and a point n on it, and construct a family of triangles with o as circumcircle and n as ninepoint center. Our proof here is of a different style than the previous one although the previous proof can be rewritten to look more like this one.

The centers of the incircle and excircles of a triangle form an orthocentric system. Putting these points together with u and v above, explain what these 4 points lie on a line and how they are situated relative to each other. Of the nine points, the three midpoints of line segments between the vertices and the orthocenter are reflections of the triangles midpoints about its ninepoint center. This ninepoint circle is also known as eulers circle, sixpoint circle. Guided discovery of the nine point circle theorem and its proof. Jan 20, 2009 history of the nine point circle volume 11 j. The ninepoint circle passes through these three midpoints. The celebrated theorem of feuerbach states that the ninepoint circle of a nonequilateral triangle istangent to both itsincircle and itsthree excircles. But, i havent seen a convincing proof for this fact yet. Paper open access introducing ninepoint circle to junior. Jan 23, 2016 elementary proof of the existence of a circle through nine points of a given triangle. Triangles with nine point center on the circumcircle begin with a circle, center o and a point n on it, and construct a family of triangles with o as circumcircle and n as nine point center. The radius of the nine point cirlce is r 2, where r is the circumradius radius of the circumcircle.

The theorem we are going to prove is the existence of the nine point circle, which is a circle created using nine important points of a triangle. This theorem states that the ninepoint circle just touches, without intersecting, the incircle and the three excircles of the triangle. The nine point circle owes its discovery to a group of famous mathematicians over the course of about 40 years, though it is most generally though perhaps not most fairly attributed to karl feuerbach, a german mathematician who rediscovered it in the nineteenth century however it was known even to euler dorrie, 100 great problems. The last three points are from the midpoint of each line segment from the. Can you find a condition on abc which ensures that the ninepoint circle touches bc. T his nine point circle is also known as eulers circle, sixpoint circle, feuerbachs circle, the twelvepoint circle, and many others. Foundational standards understand congruence and similarity using physical models, transparencies, or geometry software. We will prove that all nine points lie on the circle by. The ninepoint circle passes through many other significant points of a triangle as well. The nine point circle is another circle defined from a triangle.

In some triangles, some of these points may coincide. Pdf introducing ninepoint circle to junior high school students. Introducing ninepoint circle to junior high school. The center of the ninepoint circle is the midpoint of the line segment joining the orthocenter and the circumcenter, and hence lies on.

Nine point circle tkhalid august 16, 2015 abstract iamproudtopresentoneofmy. The three midpoints of the segments joining the vertices of the triangle to its orthocenter. The ninepoint circle is often also referred to as the euler or feuerbach circle. O9 oc oa ob hb hc ha h ma mc mb b c a the nine point circle in order to prove the existence of such a circle, we break the proof into three steps. The nine point circle created for that orthocentric system is the circumcircle of the original triangle. This is a continuation of the altitudes and the euler line page, towards the end of which we established existence of the euler line. A circle consists of points which are equidistant from a fixed point centre the circle is often referred to as the circumference.

A result closely associated with the nine point circle is that of the euler line, namely that the orthocentre h, centroid g, circumcentre o. In 1822 karl feuerbach proved that the nine point circle is tangent to the incircles and excircles of the triangle. Circle geometry page 1 there are a number of definitions of the parts of a circle which you must know. As a result, e, f, g, j, k, l, h, i, and t are all equal distant from n and lie on the same circle, so n is the center of the circle that has a radius of onehalf of the circumscribed circle and thus proves the. For example, there is the following fact which adds the nine point circle centre to the list of points lying on the euler line. Proving circle theorems angle in a semicircle we want to prove that the angle subtended at the circumference by a semicircle is a right angle. Those nine points are the midpoint of each side, the feet of each altitude, and the midpoints of the segments connecting the orthocenter with each vertex. The nine point circle passes through many other significant points of a triangle as well. Geometry articles, theorems, problems, and interactive. It establishes an existence of a circle passing through nine points, all of which are related to a single triangle. We can create the circle given nine distinct points on a triangle. Instructions for its creation are here, and simplified here. Three midpoints of the sides of given triangle three feet of the altitudes of. For the love of physics walter lewin may 16, 2011 duration.