Special Isocubics in the Triangle Plane
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Angle Chasing
Angle Chasing Ray Li June 12, 2017 1 Facts you should know 1. Let ABC be a triangle and extend BC past C to D: Show that \ACD = \BAC + \ABC: 2. Let ABC be a triangle with \C = 90: Show that the circumcenter is the midpoint of AB: 3. Let ABC be a triangle with orthocenter H and feet of the altitudes D; E; F . Prove that H is the incenter of 4DEF . 4. Let ABC be a triangle with orthocenter H and feet of the altitudes D; E; F . Prove (i) that A; E; F; H lie on a circle diameter AH and (ii) that B; E; F; C lie on a circle with diameter BC. 5. Let ABC be a triangle with circumcenter O and orthocenter H: Show that \BAH = \CAO: 6. Let ABC be a triangle with circumcenter O and orthocenter H and let AH and AO meet the circumcircle at D and E, respectively. Show (i) that H and D are symmetric with respect to BC; and (ii) that H and E are symmetric with respect to the midpoint BC: 7. Let ABC be a triangle with altitudes AD; BE; and CF: Let M be the midpoint of side BC. Show that ME and MF are tangent to the circumcircle of AEF: 8. Let ABC be a triangle with incenter I, A-excenter Ia, and D the midpoint of arc BC not containing A on the circumcircle. Show that DI = DIa = DB = DC: 9. Let ABC be a triangle with incenter I and D the midpoint of arc BC not containing A on the circumcircle. -
Relationships Between Six Excircles
Sangaku Journal of Mathematics (SJM) c SJM ISSN 2534-9562 Volume 3 (2019), pp.73-90 Received 20 August 2019. Published on-line 30 September 2019 web: http://www.sangaku-journal.eu/ c The Author(s) This article is published with open access1. Relationships Between Six Excircles Stanley Rabinowitz 545 Elm St Unit 1, Milford, New Hampshire 03055, USA e-mail: [email protected] web: http://www.StanleyRabinowitz.com/ Abstract. If P is a point inside 4ABC, then the cevians through P divide 4ABC into smaller triangles of various sizes. We give theorems about the rela- tionship between the radii of certain excircles of some of these triangles. Keywords. Euclidean geometry, triangle geometry, excircles, exradii, cevians. Mathematics Subject Classification (2010). 51M04. 1. Introduction Let P be any point inside a triangle ABC. The cevians through P divide 4ABC into six smaller triangles. In a previous paper [5], we found relationships between the radii of the circles inscribed in these triangles. For example, if P is at the orthocenter H, as shown in Figure 1, then we found that r1r3r5 = r2r4r6, where the ri are radii of the incircles as shown in the figure. arXiv:1910.00418v1 [math.HO] 28 Sep 2019 Figure 1. r1r3r5 = r2r4r6 In this paper, we will find similar results using excircles instead of incircles. When the cevians through a point P interior to a triangle ABC are drawn, many smaller 1This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. -
Point of Concurrency the Three Perpendicular Bisectors of a Triangle Intersect at a Single Point
3.1 Special Segments and Centers of Classifications of Triangles: Triangles By Side: 1. Equilateral: A triangle with three I CAN... congruent sides. Define and recognize perpendicular bisectors, angle bisectors, medians, 2. Isosceles: A triangle with at least and altitudes. two congruent sides. 3. Scalene: A triangle with three sides Define and recognize points of having different lengths. (no sides are concurrency. congruent) Jul 249:36 AM Jul 249:36 AM Classifications of Triangles: Special Segments and Centers in Triangles By angle A Perpendicular Bisector is a segment or line 1. Acute: A triangle with three acute that passes through the midpoint of a side and angles. is perpendicular to that side. 2. Obtuse: A triangle with one obtuse angle. 3. Right: A triangle with one right angle 4. Equiangular: A triangle with three congruent angles Jul 249:36 AM Jul 249:36 AM Point of Concurrency The three perpendicular bisectors of a triangle intersect at a single point. Two lines intersect at a point. The point of When three or more lines intersect at the concurrency of the same point, it is called a "Point of perpendicular Concurrency." bisectors is called the circumcenter. Jul 249:36 AM Jul 249:36 AM 1 Circumcenter Properties An angle bisector is a segment that divides 1. The circumcenter is an angle into two congruent angles. the center of the circumscribed circle. BD is an angle bisector. 2. The circumcenter is equidistant to each of the triangles vertices. m∠ABD= m∠DBC Jul 249:36 AM Jul 249:36 AM The three angle bisectors of a triangle Incenter properties intersect at a single point. -
Median and Altitude of a Triangle Goal: • to Use Properties of the Medians
Median and Altitude of a Triangle Goal: • To use properties of the medians of a triangle. • To use properties of the altitudes of a triangle. Median of a Triangle Median of a Triangle – a segment whose endpoints are the vertex of a triangle and the midpoint of the opposite side. Vertex Median Median of an Obtuse Triangle A D Point of concurrency “P” or centroid E P C B F Medians of a Triangle The medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side. A D If P is the centroid of ABC, then AP=2 AF E 3 P C CP=22CE and BP= BD B F 33 Example - Medians of a Triangle P is the centroid of ABC. PF 5 Find AF and AP A D E P C B F 5 Median of an Acute Triangle A Point of concurrency “P” or centroid E D P C B F Median of a Right Triangle A F Point of concurrency “P” or centroid E P C B D The three medians of an obtuse, acute, and a right triangle always meet inside the triangle. Altitude of a Triangle Altitude of a triangle – the perpendicular segment from the vertex to the opposite side or to the line that contains the opposite side A altitude C B Altitude of an Acute Triangle A Point of concurrency “P” or orthocenter P C B The point of concurrency called the orthocenter lies inside the triangle. Altitude of a Right Triangle The two legs are the altitudes A The point of concurrency called the orthocenter lies on the triangle. -
What's Your Altitude?
TImath.com Geometry Hey, Ortho! What’s Your Altitude? Time Required ID: 11484 45 minutes Activity Overview In this activity, students will explore the altitudes of a triangle. They will discover that the altitude can be inside, outside, or a side of the triangle. Then, students will discover that the altitudes are concurrent. The point of concurrency is the orthocenter. They should also discover the relationship between the type of triangle and the location of the point of concurrency. Finally, students will discover properties of the orthocenter in equilateral triangles. Topic: Triangle & Their Centers • Altitudes of a Triangle • Orthocenter Teacher Preparation and Notes • This activity was written to be explored with the TI-Nspire. • The answers to all problems except Exercise 8 can be answered in the TI-Nspire document. • This is an introductory activity in which students will need to know how to change between pages, construct triangles, grab and move points, measure lengths, and construct the perpendicular bisector. • The multiple choice items are self-check, and students can check them by pressing / + S. • To download the student and solution TI-Nspire documents (.tns files) and student worksheet, go to education.ti.com/exchange and enter “11484” in the quick search box. Associated Materials • HeyOrthoWhatsYourAltitude_Student.doc • HeyOrthoWhatsYourAltitude.tns • HeyOrthoWhatsYourAltitude_Soln.tns Suggested Related Activities To download any activity listed, go to education.ti.com/exchange and enter the number in the quick search box. • Centroid and Orthocenter (TI-84 Plus family) — 4618 • Exploring the Orthocenter of a Triangle (TI-84 Plus family) — 6863 • The Orthocenter of a Triangle (TI-89 Titanium) — 4597 ©2010 Texas Instruments Incorporated Teacher Page Hey, Ortho! What’s Your Altitude TImath.com Geometry Problem 1 – Exploring the Altitude of a Triangle On page 1.2, students should define the altitude of a triangle using their textbook or other source. -
Delaunay Triangulations of Points on Circles Arxiv:1803.11436V1 [Cs.CG
Delaunay Triangulations of Points on Circles∗ Vincent Despr´e † Olivier Devillers† Hugo Parlier‡ Jean-Marc Schlenker‡ April 2, 2018 Abstract Delaunay triangulations of a point set in the Euclidean plane are ubiq- uitous in a number of computational sciences, including computational geometry. Delaunay triangulations are not well defined as soon as 4 or more points are concyclic but since it is not a generic situation, this difficulty is usually handled by using a (symbolic or explicit) perturbation. As an alter- native, we propose to define a canonical triangulation for a set of concyclic points by using a max-min angle characterization of Delaunay triangulations. This point of view leads to a well defined and unique triangulation as long as there are no symmetric quadruples of points. This unique triangulation can be computed in quasi-linear time by a very simple algorithm. 1 Introduction Let P be a set of points in the Euclidean plane. If we assume that P is in general position and in particular do not contain 4 concyclic points, then the Delaunay triangulation DT (P ) is the unique triangulation over P such that the (open) circumdisk of each triangle is empty. DT (P ) has a number of interesting properties. The one that we focus on is called the max-min angle property. For a given triangulation τ, let A(τ) be the list of all the angles of τ sorted from smallest to largest. DT (P ) is the triangulation which maximizes A(τ) for the lexicographical order [4,7] . In dimension 2 and for points in general position, this max-min angle arXiv:1803.11436v1 [cs.CG] 30 Mar 2018 property characterizes Delaunay triangulations and highlights one of their most ∗This work was supported by the ANR/FNR project SoS, INTER/ANR/16/11554412/SoS, ANR-17-CE40-0033. -
Generalized Gergonne and Nagel Points
Generalized Gergonne and Nagel Points B. Odehnal June 16, 2009 Abstract In this paper we show that the Gergonne point G of a triangle ∆ in the Euclidean plane can in fact be seen from the more general point of view, i.e., from the viewpoint of projective geometry. So it turns out that there are up to four Gergonne points associated with ∆. The Gergonne and Nagel point are isotomic conjugates of each other and thus we find up to four Nagel points associated with a generic triangle. We reformulate the problems in a more general setting and illustrate the different appearances of Gergonne points in different affine geometries. Darboux’s cubic can also be found in the more general setting and finally a projective version of Feuerbach’s circle appears. Mathematics Subject Classification (2000): 51M04, 51M05, 51B20 Key Words: triangle, incenter, excenters, Gergonne point, Brianchon’s theorem, Nagel point, isotomic conjugate, Darboux’s cubic, Feuerbach’s nine point circle. 1 Introduction Assume ∆ = {A,B,C} is a triangle with vertices A, B, and C and the incircle i. Gergonne’s point is the locus of concurrency of the three cevians connecting the contact points of i and ∆ with the opposite vertices. The Gergonne point, which can also be labelled with X7 according to [10, 11] has frequently attracted mathematician’s interest. Some generalizations have been given. So, for example, one can replace the incircle with a concentric 1 circle and replace the contact points of the incircle and the sides of ∆ with their reflections with regard to the incenter as done in [2]. -
Cevians, Symmedians, and Excircles Cevian Cevian Triangle & Circle
10/5/2011 Cevians, Symmedians, and Excircles MA 341 – Topics in Geometry Lecture 16 Cevian A cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension). B cevian C A D 05-Oct-2011 MA 341 001 2 Cevian Triangle & Circle • Pick P in the interior of ∆ABC • Draw cevians from each vertex through P to the opposite side • Gives set of three intersecting cevians AA’, BB’, and CC’ with respect to that point. • The triangle ∆A’B’C’ is known as the cevian triangle of ∆ABC with respect to P • Circumcircle of ∆A’B’C’ is known as the evian circle with respect to P. 05-Oct-2011 MA 341 001 3 1 10/5/2011 Cevian circle Cevian triangle 05-Oct-2011 MA 341 001 4 Cevians In ∆ABC examples of cevians are: medians – cevian point = G perpendicular bisectors – cevian point = O angle bisectors – cevian point = I (incenter) altitudes – cevian point = H Ceva’s Theorem deals with concurrence of any set of cevians. 05-Oct-2011 MA 341 001 5 Gergonne Point In ∆ABC find the incircle and points of tangency of incircle with sides of ∆ABC. Known as contact triangle 05-Oct-2011 MA 341 001 6 2 10/5/2011 Gergonne Point These cevians are concurrent! Why? Recall that AE=AF, BD=BF, and CD=CE Ge 05-Oct-2011 MA 341 001 7 Gergonne Point The point is called the Gergonne point, Ge. Ge 05-Oct-2011 MA 341 001 8 Gergonne Point Draw lines parallel to sides of contact triangle through Ge. -
Angles in a Circle and Cyclic Quadrilateral MODULE - 3 Geometry
Angles in a Circle and Cyclic Quadrilateral MODULE - 3 Geometry 16 Notes ANGLES IN A CIRCLE AND CYCLIC QUADRILATERAL You must have measured the angles between two straight lines. Let us now study the angles made by arcs and chords in a circle and a cyclic quadrilateral. OBJECTIVES After studying this lesson, you will be able to • verify that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle; • prove that angles in the same segment of a circle are equal; • cite examples of concyclic points; • define cyclic quadrilterals; • prove that sum of the opposite angles of a cyclic quadrilateral is 180o; • use properties of cyclic qudrilateral; • solve problems based on Theorems (proved) and solve other numerical problems based on verified properties; • use results of other theorems in solving problems. EXPECTED BACKGROUND KNOWLEDGE • Angles of a triangle • Arc, chord and circumference of a circle • Quadrilateral and its types Mathematics Secondary Course 395 MODULE - 3 Angles in a Circle and Cyclic Quadrilateral Geometry 16.1 ANGLES IN A CIRCLE Central Angle. The angle made at the centre of a circle by the radii at the end points of an arc (or Notes a chord) is called the central angle or angle subtended by an arc (or chord) at the centre. In Fig. 16.1, ∠POQ is the central angle made by arc PRQ. Fig. 16.1 The length of an arc is closely associated with the central angle subtended by the arc. Let us define the “degree measure” of an arc in terms of the central angle. -
Application of Nine Point Circle Theorem
Application of Nine Point Circle Theorem Submitted by S3 PLMGS(S) Students Bianca Diva Katrina Arifin Kezia Aprilia Sanjaya Paya Lebar Methodist Girls’ School (Secondary) A project presented to the Singapore Mathematical Project Festival 2019 Singapore Mathematic Project Festival 2019 Application of the Nine-Point Circle Abstract In mathematics geometry, a nine-point circle is a circle that can be constructed from any given triangle, which passes through nine significant concyclic points defined from the triangle. These nine points come from the midpoint of each side of the triangle, the foot of each altitude, and the midpoint of the line segment from each vertex of the triangle to the orthocentre, the point where the three altitudes intersect. In this project we carried out last year, we tried to construct nine-point circles from triangulated areas of an n-sided polygon (which we call the “Original Polygon) and create another polygon by connecting the centres of the nine-point circle (which we call the “Image Polygon”). From this, we were able to find the area ratio between the areas of the original polygon and the image polygon. Two equations were found after we collected area ratios from various n-sided regular and irregular polygons. Paya Lebar Methodist Girls’ School (Secondary) 1 Singapore Mathematic Project Festival 2019 Application of the Nine-Point Circle Acknowledgement The students involved in this project would like to thank the school for the opportunity to participate in this competition. They would like to express their gratitude to the project supervisor, Ms Kok Lai Fong for her guidance in the course of preparing this project. -
3. Adam Also Needs to Know the Altitude of the Smaller Triangle Within the Sign
Lesson 3: Special Right Triangles Standard: MCC9‐12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. Essential Question: What are the relationships among the side lengths of a 30° - 60° - 90° triangle? How can I use these relationships to solve real-world problems? Discovering Special Triangles Learning Task (30-60-90 triangle) 1. Adam, a construction manager in a nearby town, needs to check the uniformity of Yield signs around the state and is checking the heights (altitudes) of the Yield signs in your locale. Adam knows that all yield signs have the shape of an equilateral triangle. Why is it sufficient for him to check just the heights (altitudes) of the signs to verify uniformity? 2. A Yield sign created in Adam’s plant is pictured above. It has the shape of an equilateral triangle with a side length of 2 feet. If you draw the altitude of the triangular sign, you split the Yield sign in half vertically. Use what you know about triangles to find the angle measures and the side lengths of the two smaller triangles formed by the altitude. a. What does an altitude of an equilateral triangle do to the triangle? b. What kinds of triangles are formed when you draw the altitude? c. What did the altitude so to the base of the triangle? d. What did the altitude do to the angle from which it was drawn? 3. Adam also needs to know the altitude of the smaller triangle within the sign. -
Nagel , Speiker, Napoleon, Torricelli
Nagel , Speiker, Napoleon, Torricelli MA 341 – Topics in Geometry Lecture 17 Centroid The point of concurrency of the three medians. 07-Oct-2011 MA 341 2 Circumcenter Point of concurrency of the three perpendicular bisectors. 07-Oct-2011 MA 341 3 Orthocenter Point of concurrency of the three altitudes. 07-Oct-2011 MA 341 4 Incenter Point of concurrency of the three angle bisectors. 07-Oct-2011 MA 341 5 Symmedian Point Point of concurrency of the three symmedians. 07-Oct-2011 MA 341 6 Gergonne Point Point of concurrency of the three segments from vertices to intangency points. 07-Oct-2011 MA 341 7 Spieker Point The Spieker point is the incenter of the medial triangle. 07-Oct-2011 MA 341 8 Nine Point Circle Center The 9 point circle center is midpoint of the Euler segment. 07-Oct-2011 MA 341 9 Mittenpunkt Point The mittenpunkt of ΔABC is the symmedian point of the excentral triangle 05-Oct-2011 MA 341 001 10 Nagel Point Nagel point = point of concurrency of cevians to points of tangency of excircles 05-Oct-2011 MA 341 001 11 Nagel Point 05-Oct-2011 MA 341 001 12 The Nagel Point Ea has the unique property of being the point on the perimeter that is exactly half way around the triangle from A. = AB+BEaa AC+CE =s Then =- =- =- =- BEaa s AB s c and CE s AC s b BE sc- a = - CEa s b 07-Oct-2011 MA 341 13 The Nagel Point Likewise CE sa- b = - AEb s c AE sb- c = - BEc s a Apply Ceva’s Theorem AE BE CE sbscsa--- cab=´´=1 --- BEcab CE AE s a s b s c 07-Oct-2011 MA 341 14 The Nagel Segment 1.