DOCSLIB.ORG

Geometry: Neutral MATH 3120, Spring 2016 Many Theorems of Geometry Are True Regardless of Which Parallel Postulate Is Used

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometry: Neutral MATH 3120, Spring 2016 Many Theorems of Geometry Are True Regardless of Which Parallel Postulate Is Used Geometry: Neutral MATH 3120, Spring 2016 Many theorems of geometry are true regardless of which parallel postulate is used. A neutral geom- etry is one in which no parallel postulate exists, and the theorems of a netural geometry are true for Euclidean and (most) non-Euclidean geomteries. Spherical geometry is a special case of Non-Euclidean geometries where the great circles on the sphere are lines. This leads to spherical trigonometry where triangles have angle measure sums greater than 180◦. While this is a non-Euclidean geometry, spherical geometry develops along a separate path where the axioms and theorems of neutral geometry do not typically apply. The axioms and theorems of netural geometry apply to Euclidean and hyperbolic geometries. The theorems below can be proven using the SMSG axioms 1 through 15. In the SMSG axiom list, Axiom 16 is the Euclidean parallel postulate. A neutral geometry assumes only the first 15 axioms of the SMSG set. Notes on notation: The SMSG axioms refer to the length or measure of line segments and the measure of angles. Thus, we will use the notation AB to describe a line segment and AB to denote its length −−! −! or measure. We refer to the angle formed by AB and AC as \BAC (with vertex A) and denote its measure as m\BAC. 1 Lines and Angles Definitions: Congruence • Segments and Angles. Two segments (or angles) are congruent if and only if their measures are equal. • Polygons. Two polygons are congruent if and only if there exists a one-to-one correspondence between their vertices such that all their corresponding sides (line sgements) and all their corre- sponding angles are congruent. Theorem 1 Congruence relations are equivalance relations. Specifically, they are symmetric, reflexive and transitive. y Definitions: Points and Lines • Collinear Two points are collinear if they lie on the same line. • Intersecting Lines. Two lines intersect if there exists a point that is on both lines. • Parallel Lines. Parallel lines are two lines in the same plan which do not intersect. • Concurrent Lines. Concurrent lines are three or more coplanar lines that have a point in common. • Between. A point B is between A and C if B 2 AC but B 6= A and B 6= C. • Convex. A set (or polygon) P is convex if A; B 2 P =) AB 2 P . • Segment. A segment AB is the set of points A, B and all of the points between A and B. 1 −−! ! • Ray.A ray AB (also called a half-line) is a subset of the line AB that contains a given point ! A and all the points C 2 AB such that A is not between C and B. The point A is called the endpoint of the ray. • Angle. An angle is the union of two noncollinear rays which have the same endpoint. • Midpoint. The midpoint C of AB lies on AB such that AC = CB. −−! • Angle Bisector. The the ray BD is the angle bisector of \ABC if m\ABD = m\DBC. Note that angles must have measure ≤ 180◦, so reflex angles are ruled out of consideration. Theorem 2 Every line segment has exactly one midpoint. Every angle has exactly one bisector. y Theorem 3 Supplements and complements of the same angles are congruent. y Theorem 4 Vertical angles are congruent. y Theorem 5 Pasch's Axiom. Given a line that contains no vertex of a triangle, if that line intersects one side of the triangle, it must intersect another. y −−! −! −−! −−! Theorem 6 Crossbar Theorem. If AD is between AC and AB, then AD intersects BC. 2 Triangles Triangles exist in a neutral geometry and have many familiar properties from Euclidean geometry. The lack of a parallel postuatle results in triangles whose angle sum is less than or equal to 180◦. The fact that many common notions about triangles are true in netural goemetries helps explain why the ancients misunderstood Euclid's parallel postulate and its consequences. Theorem 7 Isosceles Triangle Theorem. If two sides of a triangle are congruent, then the angles opposite these sides are congruent.y Theorem 8 A point is on the perpendicular bisector of a line segment if and only if it is equidistant from the endpoints of the line segment. Definitions: Triangles • Triangle. A triangle is the union of three line segments determined by three non-collinear points. • Exterior Angle. Any angle that is both supplementary and adjacent to an angle of a triangle (or polygon) is an exterior angle of the triangle (or polygon). • Scalene Triangle. The sides of a scalene triangle all have different lengths. Theorem 9 Each exterior angle of a triangle is greater in measure than either of the nonadjacent interior angles of the triangle. Theorem 10 ASA Triangle Congruence. If two angles and the included side of one triangle are congruent, repsectively, to two angles and the included side of a second triangle, then the triangles are congruent. Theorem 11 Converse of Isosceles Triangle Theorem. If two angles of a triangle are congruent, then the sides opposite these angles are congruent. 2 Theorem 12 AAS Triangle Congruence. If the vertices of two triangles are in one-to-one cor- respondence such that two angles and the side opposite in one triangle are congruent to their corresponding parts of a second triangle, then the triangles are congruent. Theorem 13 SASAS Quadrilateral Congruence. If the vertices of two convex quadrilaterals are in one-to-one correspondence such that three sides and the two included interior angles of one quadrilateral are congrent to the corresponding parts of a second quadrilateral, then the quadrilaterals are congruent. Theorem 14 Inverse of Isosceles Triangle Theorem. If two sides of a triangle are not congruent, then angles opposite those sides are not congurent, and the larger angle is opposite the larger side. Corollary The largest (smallest) angle of a triangle is opposite the largest (smallest) side. Theorem 15 Triangle Inequality. The sum of the measures of two sides of a triangle is greater than the measure of the third side. Theorem 16 Hinge Theorem. If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. (Note that this is Euclid's Proposition 24 and is sometimes called the \Open Mouth Theorem.") Theorem 17 SSS Triangle Congruence. If the sides of one triangle are congruent to corresponding sides of another triangle, the triangles are congruent. Theorem 18 The angles of a scalene triangle all have different measures, and the angle with the largest measure is opposite the side with the largest length. Theorem 19 Prove or disprove: The analog of the SSS congruence condition for triangles is true for quadrilaterals (SSSS). Theorem 20 Two right triangles are congruent if the hypotenuse and one leg of one triangle are congruent to corresponding sides of the other. Theorem 21 Equilateral trianges are equiangular. Theorem 22 Alternate Interior Angle Theorem. If two lines are intersected by a transversal such that a pair of alternate interior angles is congreuent, then the lines are parallel. Corollary 22.1 Two lines perpendicular to the same line are parallel. Corollary 22.2 If two lines are intersected by a transversal such that a pair of corresponding angles is congreuent, then the lines are parallel. 3 Results of Euclid's Parallel Postulate Euclid's Fifth Postulate says: \That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." Awkward, and it doesn't mention a parallel condition at all. Modern texts use the Playfair Postulate which is equivalent. The Scottish Mathematician John Playfair reformulated Eculid's Fifth Postulate in the late 1700's, and the SMSG axiom set (Axiom 16) uses a version of the Playfair Postulate: \Through a given external point there is at most one line parallel to a given line." We will use SMSG Axiom as Playfair's Postulate and assume the following theorem is true: 3 Theorem 23 Euclid's parallel postulate is equivalent to Playfair's Postulate and SMSG Axiom 16. Theorem 24 The Playfair Postulate (and Euclid's parallel postulate) is equivalent to the Converse of the Alternate Interior Angle theorem (Theorem 23). Theorem 25 Euclid's parallel postulate is equivalent to the statement: If a line intersects one of two parallel lines, then it intersects the other. Theorem 26 Euclid's parallel postulate is equivalent to the statement: If a line is perpendicular to one of two parallel lines, then it is perpedicular to the other. Theorem 27 Wallis's Postulate. Euclid's parallel postulate is equivalent to the statement: Given a 4ABC and any line segment DE, there exists a triangle having DE as one of its sides that is similar to 4ABC. 4 The Saccheri-Legendre Theorem Theorem 28 The angle sum of any triangle is less than or equal to 180◦. Lemma 28.1 The sum of the measures of any two angles of a triangle is less than 180◦. Lemma 28.2 For any 4ABC there exists 4A1B1C1 having the same angle sum as 4ABC but 1 where m\A1 = 2 m\A. Corollary 28.3 The angle sum of any convex quadrilateral is less than or equal to 360◦. An important result of the angle sum being no more than 180◦ is the following theorem: Theorem 29: Unique Perpendicular Given a line l and any point P not on l, there exists a unique line m such that P 2 m and m ? l.
Load more

Recommended publications
  • Geometry Notes G.2 Parallel Lines, Transversals, Angles Mrs. Grieser Name: Date
    Geometry Notes G.2 Parallel Lines, Transversals, Angles Mrs. Grieser Name: ________________________________________ Date: ______________ Block: ________ Identifying Pairs of Lines and Angles Lines that intersect are coplanar Lines that do not intersect o are parallel (coplanar) OR o are skew (not coplanar) In the figure, name: o intersecting lines:___________ o parallel lines:_______________ o skew lines:________________ o parallel planes:_____________ Example: Think of each segment in the figure as part of a line. Find: Line(s) parallel to CD and containing point A _________ Line(s) skew to and containing point A _________ Line(s) perpendicular to and containing point A _________ Plane(s) parallel to plane EFG and containing point A _________ Parallel and Perpendicular Lines If two lines are in the same plane, then they are either parallel or intersect a point. There exist how many lines through a point not on a line? __________ Only __________ of them is parallel to the line. Parallel Postulate If there is a line and a point not on a line, then there is exactly one line through the point parallel to the given line. Perpendicular Postulate If there is a line and a point not on a line, then there is exactly one line through the point parallel to the given line. Angles and Transversals Interior angles are on the INSIDE of the two lines Exterior angles are on the OUTSIDE of the two lines Alternate angles are on EITHER SIDE of the transversal Consecutive angles are on the SAME SIDE of the transversal Corresponding angles are in the same position on each of the two lines Alternate interior angles lie on either side of the transversal inside the two lines Alternate exterior angles lie on either side of the transversal outside the two lines Consecutive interior angles lie on the same side of the transversal inside the two lines (same side interior) Geometry Notes G.2 Parallel Lines, Transversals, Angles Mrs.
    [Show full text]
  • Proofs with Perpendicular Lines
    3.4 Proofs with Perpendicular Lines EEssentialssential QQuestionuestion What conjectures can you make about perpendicular lines? Writing Conjectures Work with a partner. Fold a piece of paper D in half twice. Label points on the two creases, as shown. a. Write a conjecture about AB— and CD — . Justify your conjecture. b. Write a conjecture about AO— and OB — . AOB Justify your conjecture. C Exploring a Segment Bisector Work with a partner. Fold and crease a piece A of paper, as shown. Label the ends of the crease as A and B. a. Fold the paper again so that point A coincides with point B. Crease the paper on that fold. b. Unfold the paper and examine the four angles formed by the two creases. What can you conclude about the four angles? B Writing a Conjecture CONSTRUCTING Work with a partner. VIABLE a. Draw AB — , as shown. A ARGUMENTS b. Draw an arc with center A on each To be prof cient in math, side of AB — . Using the same compass you need to make setting, draw an arc with center B conjectures and build a on each side of AB— . Label the C O D logical progression of intersections of the arcs C and D. statements to explore the c. Draw CD — . Label its intersection truth of your conjectures. — with AB as O. Write a conjecture B about the resulting diagram. Justify your conjecture. CCommunicateommunicate YourYour AnswerAnswer 4. What conjectures can you make about perpendicular lines? 5. In Exploration 3, f nd AO and OB when AB = 4 units.
    [Show full text]
  • Projective Geometry: a Short Introduction
    Projective Geometry: A Short Introduction Lecture Notes Edmond Boyer Master MOSIG Introduction to Projective Geometry Contents 1 Introduction 2 1.1 Objective . .2 1.2 Historical Background . .3 1.3 Bibliography . .4 2 Projective Spaces 5 2.1 Definitions . .5 2.2 Properties . .8 2.3 The hyperplane at infinity . 12 3 The projective line 13 3.1 Introduction . 13 3.2 Projective transformation of P1 ................... 14 3.3 The cross-ratio . 14 4 The projective plane 17 4.1 Points and lines . 17 4.2 Line at infinity . 18 4.3 Homographies . 19 4.4 Conics . 20 4.5 Affine transformations . 22 4.6 Euclidean transformations . 22 4.7 Particular transformations . 24 4.8 Transformation hierarchy . 25 Grenoble Universities 1 Master MOSIG Introduction to Projective Geometry Chapter 1 Introduction 1.1 Objective The objective of this course is to give basic notions and intuitions on projective geometry. The interest of projective geometry arises in several visual comput- ing domains, in particular computer vision modelling and computer graphics. It provides a mathematical formalism to describe the geometry of cameras and the associated transformations, hence enabling the design of computational ap- proaches that manipulates 2D projections of 3D objects. In that respect, a fundamental aspect is the fact that objects at infinity can be represented and manipulated with projective geometry and this in contrast to the Euclidean geometry. This allows perspective deformations to be represented as projective transformations. Figure 1.1: Example of perspective deformation or 2D projective transforma- tion. Another argument is that Euclidean geometry is sometimes difficult to use in algorithms, with particular cases arising from non-generic situations (e.g.
    [Show full text]
  • Theorems and Postulates
    Theorems and Postulates JJJG Given AB and a number r between 0 and 180, there Postulate 1-A is exactly one JrayJJG with endpoint A , extending on Protractor Postulate either side of AB , such that the measure of the angle formed is r . ∠A is a right angle if mA∠ is 90. Definition of Right, Acute ∠A is an acute angle if mA∠ is less than 90. and Obtuse Angles ∠A is an obtuse angle if mA∠ is greater than 90 and less than 180. Postulate 1-B If R is in the interior of ∠PQS , then mPQRmRQSmPQS∠ +∠ =∠ . Angle Addition If mP∠+∠=∠QRmRQSmPQS, then R is in the interior of ∠PQS. Vertical angles are congruent. The sum of the measures of the angles in a linear pair is 180˚. The sum of the measures of complementary angles is 90˚. Two points on a line can be paired with real numbers so that, given Postulate 2-A any two points R and S on the line, R corresponds to zero, and S Ruler corresponds to a positive number. Point R could be paired with 0, and S could be paired with 10. R S -2 -1 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 10 Postulate 2-B If N is between M and P, then MN + NP = MP. Segment Addition Conversely, if MN + NP = MP, then N is between M and P. Theorem 2-A In a right triangle, the sum of the squares of the measures Pythagorean of the legs equals the square of the measure of the Theorem hypotenuse.
    [Show full text]
  • Parallel Lines and Transversals
    5.5 Parallel Lines and Transversals How can you use STATES properties of parallel lines to solve real-life problems? STANDARDS MA.8.G.2.2 1 ACTIVITY: A Property of Parallel Lines Work with a partner. ● Talk about what it means for two lines 12 1 cm 56789 to be parallel. Decide on a strategy for 1234 drawing two parallel lines. 01 91 8 9 9 9 9 10 11 12 13 10 10 10 10 ● Use your strategy to carefully draw 7 67 two lines that are parallel. 14 15 5 16 17 18 19 20 3421 22 23 2 24 25 26 1 ● Now, draw a third line 27 28 in. parallel that intersects the two 29 30 lines parallel lines. This line is called a transversal. 2 1 3 4 6 ● The two parallel lines and 5 7 8 the transversal form eight angles. Which of these angles have equal measures? transversal Explain your reasoning. 2 ACTIVITY: Creating Parallel Lines Work with a partner. a. If you were building the house in the photograph, how could you make sure that the studs are parallel to each other? b. Identify sets of parallel lines and transversals in the Studs photograph. 212 Chapter 5 Angles and Similarity 3 ACTIVITY: Indirect Measurement Work with a partner. F a. Use the fact that two rays from the Sun are parallel to explain why △ABC and △DEF are similar. b. Explain how to use similar triangles to fi nd the height of the fl agpole. x ft Sun’s ray C Sun’s ray 5 ft AB3 ft DE36 ft 4.
    [Show full text]
  • Definition Concurrent Lines Are Lines That Intersect in a Single Point. 1. Theorem 128: the Perpendicular Bisectors of the Sides
    14.3 Notes Thursday, April 23, 2009 12:49 PM Definition 1. Concurrent lines are lines that intersect in a single point. j k m Theorem 128: The perpendicular bisectors of the sides of a triangle are concurrent at a point that is equidistant from the vertices of the triangle. This point is called the circumcenter of the triangle. D E F Theorem 129: The bisectors of the angles of a triangle are concurrent at a point that is equidistant from the sides of the triangle. This point is called the incenter of the triangle. A B Notes Page 1 C A B C Theorem 130: The lines containing the altitudes of a triangle are concurrent. This point is called the orthocenter of the triangle. A B C Theorem 131: The medians of a triangle are concurrent at a point that is 2/3 of the way from any vertex of the triangle to the midpoint of the opposite side. This point is called the centroid of the of the triangle. Example 1: Construct the incenter of ABC A B C Notes Page 2 14.4 Notes Friday, April 24, 2009 1:10 PM Examples 1-3 on page 670 1. Construct an angle whose measure is equal to 2A - B. A B 2. Construct the tangent to circle P at point A. P A 3. Construct a tangent to circle O from point P. Notes Page 3 3. Construct a tangent to circle O from point P. O P Notes Page 4 14.5 notes Tuesday, April 28, 2009 8:26 AM Constructions 9, 10, 11 Geometric mean Notes Page 5 14.6 Notes Tuesday, April 28, 2009 9:54 AM Construct: ABC, given {a, ha, B} a Ha B A b c B C a Notes Page 6 14.1 Notes Tuesday, April 28, 2009 10:01 AM Definition: A locus is a set consisting of all points, and only the points, that satisfy specific conditions.
    [Show full text]
  • Downloaded from Bookstore.Ams.Org 30-60-90 Triangle, 190, 233 36-72
    Index 30-60-90 triangle, 190, 233 intersects interior of a side, 144 36-72-72 triangle, 226 to the base of an isosceles triangle, 145 360 theorem, 96, 97 to the hypotenuse, 144 45-45-90 triangle, 190, 233 to the longest side, 144 60-60-60 triangle, 189 Amtrak model, 29 and (logical conjunction), 385 AA congruence theorem for asymptotic angle, 83 triangles, 353 acute, 88 AA similarity theorem, 216 included between two sides, 104 AAA congruence theorem in hyperbolic inscribed in a semicircle, 257 geometry, 338 inscribed in an arc, 257 AAA construction theorem, 191 obtuse, 88 AAASA congruence, 197, 354 of a polygon, 156 AAS congruence theorem, 119 of a triangle, 103 AASAS congruence, 179 of an asymptotic triangle, 351 ABCD property of rigid motions, 441 on a side of a line, 149 absolute value, 434 opposite a side, 104 acute angle, 88 proper, 84 acute triangle, 105 right, 88 adapted coordinate function, 72 straight, 84 adjacency lemma, 98 zero, 84 adjacent angles, 90, 91 angle addition theorem, 90 adjacent edges of a polygon, 156 angle bisector, 100, 147 adjacent interior angle, 113 angle bisector concurrence theorem, 268 admissible decomposition, 201 angle bisector proportion theorem, 219 algebraic number, 317 angle bisector theorem, 147 all-or-nothing theorem, 333 converse, 149 alternate interior angles, 150 angle construction theorem, 88 alternate interior angles postulate, 323 angle criterion for convexity, 160 alternate interior angles theorem, 150 angle measure, 54, 85 converse, 185, 323 between two lines, 357 altitude concurrence theorem,
    [Show full text]
  • Triangles and Transversals Triangles
    Triangles and Transversals Triangles A three-sided polygon. Symbol → ▵ You name it ▵ ABC. Total Angle Sum of a Triangle The interior angles of a triangle add up to 180 degrees. Symbol = Acute Triangle A triangle that contains only angles that are less than 90 degrees. Obtuse Triangle A triangle with one angle greater than 90 degrees (an obtuse angle). Right Triangle A triangle with one right angle (90 degrees). Vertex The common endpoint of two or more rays or line segments. Complementary Angles Two angles whose measures have a sum of 90 degrees. Supplementary Angles Two angles whose measures have a sum of 180 degrees. Perpendicular Lines Lines that intersect to form a right angle (90 degrees). Symbol = Parallel Lines Lines that never intersect. Arrows are used to indicate lines are parallel. Symbol = || Transversal Lines A line that cuts across two or more (usually parallel) lines. Intersect The point where two lines meet or cross. Vertical Angles Angles opposite one another at the intersection of two lines. Vertical angles have the same angle measurements. Interior Angles An angle inside a shape. Exterior Angles Angles outside of a shape. Alternate Interior Angles The pairs of angles on opposite sides of the transversal but inside the two lines. Alternate Exterior Angles Each pair of these angles are outside the lines, and on opposite sides of the transversal. Corresponding Angles The angles in matching corners. Reflexive Angles An angle whose measure is greater than 180 degrees and less that 360 degrees. Straight Angles An angle that measures exactly 180 degrees. Adjacent Angles Angles with common side and common vertex without overlapping.
    [Show full text]
  • Read Book Advanced Euclidean Geometry Ebook
    ADVANCED EUCLIDEAN GEOMETRY PDF, EPUB, EBOOK Roger A. Johnson | 336 pages | 30 Nov 2007 | Dover Publications Inc. | 9780486462370 | English | New York, United States Advanced Euclidean Geometry PDF Book As P approaches nearer to A , r passes through all values from one to zero; as P passes through A , and moves toward B, r becomes zero and then passes through all negative values, becoming —1 at the mid-point of AB. Uh-oh, it looks like your Internet Explorer is out of date. In Elements Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed angle theorem Intercept theorem Pons asinorum Pythagorean theorem Thales's theorem Theorem of the gnomon. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross. Calculus Real analysis Complex analysis Differential equations Functional analysis Harmonic analysis. This article needs attention from an expert in mathematics. Facebook Twitter. On any line there is one and only one point at infinity. This may be formulated and proved algebraically:. When we have occasion to deal with a geometric quantity that may be regarded as measurable in either of two directions, it is often convenient to regard measurements in one of these directions as positive, the other as negative. Logical questions thus become completely independent of empirical or psychological questions For example, proposition I. This volume serves as an extension of high school-level studies of geometry and algebra, and He was formerly professor of mathematics education and dean of the School of Education at The City College of the City University of New York, where he spent the previous 40 years.
    [Show full text]
  • Understand the Principles and Properties of Axiomatic (Synthetic
    Michael Bonomi Understand the principles and properties of axiomatic (synthetic) geometries (0016) Euclidean Geometry: To understand this part of the CST I decided to start off with the geometry we know the most and that is Euclidean: − Euclidean geometry is a geometry that is based on axioms and postulates − Axioms are accepted assumptions without proofs − In Euclidean geometry there are 5 axioms which the rest of geometry is based on − Everybody had no problems with them except for the 5 axiom the parallel postulate − This axiom was that there is only one unique line through a point that is parallel to another line − Most of the geometry can be proven without the parallel postulate − If you do not assume this postulate, then you can only prove that the angle measurements of right triangle are ≤ 180° Hyperbolic Geometry: − We will look at the Poincare model − This model consists of points on the interior of a circle with a radius of one − The lines consist of arcs and intersect our circle at 90° − Angles are defined by angles between the tangent lines drawn between the curves at the point of intersection − If two lines do not intersect within the circle, then they are parallel − Two points on a line in hyperbolic geometry is a line segment − The angle measure of a triangle in hyperbolic geometry < 180° Projective Geometry: − This is the geometry that deals with projecting images from one plane to another this can be like projecting a shadow − This picture shows the basics of Projective geometry − The geometry does not preserve length
    [Show full text]
  • The Stoics and the Practical: a Roman Reply to Aristotle
    DePaul University Via Sapientiae College of Liberal Arts & Social Sciences Theses and Dissertations College of Liberal Arts and Social Sciences 8-2013 The Stoics and the practical: a Roman reply to Aristotle Robin Weiss DePaul University, [email protected] Follow this and additional works at: https://via.library.depaul.edu/etd Recommended Citation Weiss, Robin, "The Stoics and the practical: a Roman reply to Aristotle" (2013). College of Liberal Arts & Social Sciences Theses and Dissertations. 143. https://via.library.depaul.edu/etd/143 This Thesis is brought to you for free and open access by the College of Liberal Arts and Social Sciences at Via Sapientiae. It has been accepted for inclusion in College of Liberal Arts & Social Sciences Theses and Dissertations by an authorized administrator of Via Sapientiae. For more information, please contact [email protected]. THE STOICS AND THE PRACTICAL: A ROMAN REPLY TO ARISTOTLE A Thesis Presented in Partial Fulfillment of the Degree of Doctor of Philosophy August, 2013 BY Robin Weiss Department of Philosophy College of Liberal Arts and Social Sciences DePaul University Chicago, IL - TABLE OF CONTENTS - Introduction……………………..............................................................................................................p.i Chapter One: Practical Knowledge and its Others Technê and Natural Philosophy…………………………….....……..……………………………….....p. 1 Virtue and technical expertise conflated – subsequently distinguished in Plato – ethical knowledge contrasted with that of nature in
    [Show full text]
  • The Dual Theorem Concerning Aubert Line
    The Dual Theorem concerning Aubert Line Professor Ion Patrascu, National College "Buzeşti Brothers" Craiova - Romania Professor Florentin Smarandache, University of New Mexico, Gallup, USA In this article we introduce the concept of Bobillier transversal of a triangle with respect to a point in its plan; we prove the Aubert Theorem about the collinearity of the orthocenters in the triangles determined by the sides and the diagonals of a complete quadrilateral, and we obtain the Dual Theorem of this Theorem. Theorem 1 (E. Bobillier) Let 퐴퐵퐶 be a triangle and 푀 a point in the plane of the triangle so that the perpendiculars taken in 푀, and 푀퐴, 푀퐵, 푀퐶 respectively, intersect the sides 퐵퐶, 퐶퐴 and 퐴퐵 at 퐴푚, 퐵푚 and 퐶푚. Then the points 퐴푚, 퐵푚 and 퐶푚 are collinear. 퐴푚퐵 Proof We note that = 퐴푚퐶 aria (퐵푀퐴푚) (see Fig. 1). aria (퐶푀퐴푚) 1 Area (퐵푀퐴푚) = ∙ 퐵푀 ∙ 푀퐴푚 ∙ 2 sin(퐵푀퐴푚̂ ). 1 Area (퐶푀퐴푚) = ∙ 퐶푀 ∙ 푀퐴푚 ∙ 2 sin(퐶푀퐴푚̂ ). Since 1 3휋 푚(퐶푀퐴푚̂ ) = − 푚(퐴푀퐶̂ ), 2 it explains that sin(퐶푀퐴푚̂ ) = − cos(퐴푀퐶̂ ); 휋 sin(퐵푀퐴푚̂ ) = sin (퐴푀퐵̂ − ) = − cos(퐴푀퐵̂ ). 2 Therefore: 퐴푚퐵 푀퐵 ∙ cos(퐴푀퐵̂ ) = (1). 퐴푚퐶 푀퐶 ∙ cos(퐴푀퐶̂ ) In the same way, we find that: 퐵푚퐶 푀퐶 cos(퐵푀퐶̂ ) = ∙ (2); 퐵푚퐴 푀퐴 cos(퐴푀퐵̂ ) 퐶푚퐴 푀퐴 cos(퐴푀퐶̂ ) = ∙ (3). 퐶푚퐵 푀퐵 cos(퐵푀퐶̂ ) The relations (1), (2), (3), and the reciprocal Theorem of Menelaus lead to the collinearity of points 퐴푚, 퐵푚, 퐶푚. Note Bobillier's Theorem can be obtained – by converting the duality with respect to a circle – from the theorem relative to the concurrency of the heights of a triangle.
    [Show full text]