Nine point circle proof pdf

The first three points are the feet of the altitudes of our triangle with the name of d, e, and f. In 1822 karl feuerbach proved that the ninepoint circle is tangent to the incircles and excircles of the triangle. 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. If a is a right angle, where does it lie in relation to the ninepoint circle. The celebrated theorem of feuerbach states that the ninepoint circle of a nonequilateral triangle istangent to both itsincircle and itsthree excircles. 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. Thus, b and b as well as c and c are inverse images with respect to our inversion transformation. Pdf a generalization of the ninepoint circle and euler. 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.

A generalization of the ninepoint circle and euler line. 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. The radius of the ninepoint cirlce is r 2, where r is the circumradius radius of the circumcircle. In 1822 karl feuerbach proved that the nine point circle is tangent to the incircles and excircles of the triangle. The proof will use the line wy as the base of the triangle. Instructions for its creation are here, and simplified here. Geometry articles, theorems, problems, and interactive. The center of the nine point circle is the midpoint of the line segment joining the orthocenter and the circumcenter, and hence lies on.

Origin so called because the circle passes through nine points of interest. In this note, we give a simple proof of feuerbachs theorem using straightforward vector computations. 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. This paper describes a set of instructional activities that can help students discover the ninepoint circle theorem through investigation in a dynamic geometry. The theorem that states that n touches the incircle internally and the excir cles externally is due to feuerbach 1822 feu. The radius of the nine point cirlce is r 2, where r is the circumradius radius of the circumcircle. Jan 23, 2016 for the love of physics walter lewin may 16, 2011 duration. 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.

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. The ninepoint circle is often also referred to as the euler or feuerbach circle. Guided discovery of the nine point circle theorem and its proof. 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. 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. 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. 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. However, in trigonometry material of senior high school mathematics, the concept of triangle circumcircle is used for proving sine rule.

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. International journal of mathematical education in science and technology. We can create the circle given nine distinct points on a triangle. Pdf ninepoint circle, pedal circle and cevian circle quang. The radius of the ninepoint circle is half that of the circumcircle, and its centre bisects the line between the circumcentre and the orthocentre. 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. We will prove that all nine points lie on the circle by. A simple vector proof of feuerbac hs theorem michael j. But every triangle has three bases, and if we consider. Let abc be a triangle with orthocenter h and nine point center n. This theorem states that the ninepoint circle just touches, without intersecting, the incircle and the three excircles of the triangle. A circle consists of points which are equidistant from a fixed point centre the circle is often referred to as the circumference.

Three natural homoteties of the ninepoint circle forum. The ninepoint circle of a triangle is tangent to the incircle and each of the three excircles of the triangle. The nine point circle passes through many other significant points of a triangle as well. Feuerbachs theorem, including the first published proof, appears in karl. This is a continuation of the altitudes and the euler line page, towards the end of which we established existence of the euler line. Elementary proof of the existence of a circle through nine points of a given triangle. Because of these different names, there have been misunderstand among mathematicians about the ninepoint circles history. Paper open access introducing ninepoint circle to junior. Jan 20, 2009 history of the nine point circle volume 11 j. 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. The ninepoint circle satisfies several important and. The ninepoint circle passes through these three midpoints.

Introducing ninepoint circle to junior high school. The fact that the nine point circle exists at all is amazing in itself. Can you find a condition on abc which ensures that the ninepoint circle touches bc. A result closely associated with the ninepoint circle is that of the euler line, namely that the orthocentre h, centroid g, circumcentre o. On the ninepoint conic proceedings of the edinburgh. The nine point circle of a triangle is tangent to the incircle and each of the three excircles of the triangle.

Circle geometry page 1 there are a number of definitions of the parts of a circle which you must know. 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 nine point circle main concept the nine point 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. There are midpoints galore in this problem in fact, six of the nine points that we are interested in are defined as midpoints. 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. Foundational standards understand congruence and similarity using physical models, transparencies, or geometry software. The ninepoint circle is another circle defined from a triangle. 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.

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. 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. The nine point circle created for that orthocentric system is the circumcircle of the original 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. Three midpoints of the sides of given triangle three feet of the altitudes of.

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 nine point circle also known as eulers circle or feuerbachs circle of a given triangle is a circle which passes through 9 significant points. That circle which passes through the feet of the altitudes of a given triangle. This ninepoint circle is also known as eulers circle, sixpoint 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 black lines are construction of the points, the orange lines are construction of the circles. Nine point circle tkhalid august 16, 2015 abstract iamproudtopresentoneofmy. The nine point circle is another circle defined from a triangle. The last three points are from the midpoint of each line segment from the. 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. Pdf introducing ninepoint circle to junior high school students. The circle through the midpoints of the sides passes through the base points feet of the. A result closely associated with the nine point circle is that of the euler line, namely that the orthocentre h, centroid g, circumcentre o.

The next three points are created from the midpoints of each of the triangles sides g, h, i. The nine point circle is often also referred to as the euler or feuerbach circle. For the love of physics walter lewin may 16, 2011 duration. Introduction how would you draw a circle inside a triangle, touching all three sides. 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. Create the problem draw a circle, mark its centre and draw a diameter through the centre. In some triangles, some of these points may coincide. Trigonometrycircles and trianglesthe ninepoint circle. The ninepoint circle passes through many other significant points of a triangle as well. The main purpose of the paper is to present a new proof of the two celebrated theorems. The nine point circle passes through these three midpoints. Pdf the concept of circles is an ancient concept that has appeared since. Since b and c are on the 9points circle, and the 9pts circle passes.

Guided discovery of the ninepoint circle theorem and its. Ninepointcircle dictionary definition ninepointcircle. For example, there is the following fact which adds the nine point circle centre to the list of points lying on the euler line. Three natural homoteties of the ninepoint circle 211 theorem 3. The three midpoints of the segments joining the vertices of the triangle to its orthocenter. 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. The center of the ninepoint circle is the midpoint of the line segment joining the orthocenter and the circumcenter, and hence lies on. It establishes an existence of a circle passing through nine points, all of which are related to a single triangle. T his nine point circle is also known as eulers circle, sixpoint circle, feuerbachs circle, the twelvepoint circle, and many others. But, i havent seen a convincing proof for this fact yet. A radius is an interval which joins the centre to a point on the circumference. Abstractthe ninepoint circle theorem is one of the most beautiful and surprising theorems in euclidean geometry.

One of the mysterious features is the ninepoint circle. The fact that the ninepoint circle exists at all is amazing in itself. If, for instance, we consider a triangle abc with its orthocentre h as an orthic four point, any proof that shows that the nine point circle touches the inscribed or an escribed circle of the triangle abc, will, in general, also show that it touches the inscribed and escribed circles of the triangles hcb, cha and bah. Let o be the center of the circumcircle c and let n be the center of the ninepoint circle d.

Nt nl nf of the circumscribed circle and 1 radius 2 ni nj ne of the circumscribed circle. 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 ninepoint circle of a triangle is a circle going through 9 key points. 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. 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 of a triangle is a circle going through 9 key points. The black lines are construction of the points, the orange lines are construction of the circle s. The other two sides should meet at a vertex somewhere on the. Jan 23, 2016 elementary proof of the existence of a circle through nine points of a given triangle. Let o be the center of the circumcircle c and let n be the center of the nine point circle d. These three triples of points make nine in all, giving the circle its name. The centers of the incircle and excircles of a triangle form an orthocentric system.