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Introduction to the Geometry of the Triangle

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Introduction to the Geometry of the Triangle Introduction to the Geometry of the Triangle Paul Yiu Fall 2005 Department of Mathematics Florida Atlantic University Version 5.0924 September 2005 Contents 1 The Circumcircle and the Incircle 1 1.1 Preliminaries ................................. 1 1.1.1 Coordinatization of points on a line . ............ 1 1.1.2 Centers of similitude of two circles . ............ 2 1.1.3 Tangent circles . .......................... 3 1.1.4 Harmonic division .......................... 4 1.1.5 Homothety . .......................... 5 1.1.6 The power of a point with respect to a circle . ............ 6 1.2 Menelaus and Ceva theorems . ................... 7 1.2.1 Menelaus and Ceva Theorems . ................... 7 1.2.2 Desargues Theorem . ................... 8 1.3 The circumcircle and the incircle of a triangle . ............ 9 1.3.1 The circumcircle and the law of sines . ............ 9 1.3.2 The incircle and the Gergonne point . ............ 10 1.3.3 The Heron formula .......................... 14 1.3.4 The excircles and the Nagel point . ............ 16 1.4 The medial and antimedial triangles . ................... 20 1.4.1 The medial triangle, the nine-point center, and the Spieker point . 20 1.4.2 The antimedial triangle and the orthocenter . ............ 21 1.4.3 The Euler line . .......................... 22 1.5 The nine-point circle . .......................... 25 1.5.1 The Euler triangle as a midway triangle . ............ 25 1.5.2 The orthic triangle as a pedal triangle . ............ 26 1.5.3 The nine-point circle . ................... 27 1.6 The OI-line ................................. 31 1.6.1 The homothetic center of the intouch and excentral triangles . 31 1.6.2 The centers of similitude of the circumcircle and the incircle . 32 1.6.3 Reflection of I in O ......................... 33 1.6.4 Orthocenter of intouch triangle . ................... 34 1.6.5 Centroids of the excentral and intouch triangles........... 35 1.6.6 Mixtilinear incircles . ................... 39 1.7 Euler’s formula and Steiner’s porism . ................... 41 1.7.1 Euler’s formula . .......................... 41 1.7.2 Steiner’s porism . .......................... 42 iv CONTENTS 2 Homogeneous Barycentric Coordinates 45 2.1 Barycentric coordinates with reference to a triangle . ...... 45 2.1.1 Homogeneous barycentric coordinates . ............. 45 2.1.2 Point of division of a segment . ................. 49 2.1.3 The area formula . ........................ 51 2.2 Conway’s formula . ........................ 52 2.2.1 Conway’s Notation . ........................ 52 2.2.2 Conway’s formula . ........................ 53 2.3 Equations of Straight lines . ........................ 58 2.3.1 Two-point form . ........................ 58 2.3.2 Infinite points and parallel lines . ................. 59 2.3.3 Perpendicular lines . ........................ 61 2.3.4 Intercept form: trilinear pole and polar . ............. 63 2.3.5 Intersection of two lines . ................. 65 2.4 Cevian and anticevian triangles . ................. 68 2.4.1 Cevian triangle . ........................ 68 2.4.2 Anticevian triangle . ........................ 73 2.4.3 Construction of anticevian triangle . ................. 73 2.5 Perspectivity . ............................... 76 2.5.1 Perspector and perspectrix . ................. 76 2.5.2 Triangles bounded by lines parallel to the sidelines . ...... 77 2.5.3 The cevian nest theorem . ................. 80 Chapter 1 The Circumcircle and the Incircle 1.1 Preliminaries 1.1.1 Coordinatization of points on a line Let B and C be two fixed points on a line L. Every point X on L can be coordinatized in one of several ways: = BX (1) the ratio of division t XC , (2) the absolute barycentric coordinates: an expression of X as a convex combination of B and C: X =(1− t)B + tC, which expresses for an arbitrary point P outside the line L, the vector PX as a combination of the vectors PB and PC. (3) the homogeneous barycentric coordinates: the proportion XC : BX, which are masses at B and C so that the resulting system (of two particles) has balance point at X. P B X C 2 The Circumcircle and the Incircle 1.1.2 Centers of similitude of two circles Consider two circles O(R) and I(r), whose centers O and I are at a distance d apart. Animate a point X on O(R) and construct a ray through I oppositely parallel to the ray OX to intersect the circle I(r) at a point Y . You will find that the line XY always intersects the line OI at the same point T . This we call the internal center of similitude, or simply the insimilicenter, of the two circles. It divides the segment OI in the ratio OT : TI = R : r. The absolute barycentric coordinates of P with respect to OI are R · I + r · O T = . R + r Y T T O I Y X If, on the other hand, we construct a ray through I directly parallel to the ray OX to intersect the circle I(r) at Y , the line XY always intersects OI at another point T . This is the external center of similitude, or simply the exsimilicenter, of the two circles. It divides the segment OI in the ratio OT : T I = R : −r, and has absolute barycentric coordinates R · I − r · O T = . R − r 1.1 Preliminaries 3 1.1.3 Tangent circles If two circles are tangent to each other, the line joining their centers passes through the point of tangency, which is a center of similitude of the circles. T O T I O I 4 The Circumcircle and the Incircle 1.1.4 Harmonic division Two points X and Y are said to divide two other points B and C harmonically if BX BY = − . XC YC They are harmonic conjugates of each other with respect to the segment BC. Examples 1. For two given circles, the two centers of similitude divide the centers harmonically. 2. Given triangle ABC, let the internal bisector of angle A intersect BC at X. The harmonic conjugate of X in BC is the intersection of BC with the external bisector of angle A. A B X C X 3. Let A and B be distinct points. If M is the midpoint of the segment AB,itisnot possible to find a finite point N on the line AB so that M, N divide A, B harmon- AN = − AM = −1 = − = ically. This is because NB MB , requiring AN NB BN, and AB = BN −AN =0, a contradiction. We shall agree to say that if M and N divide A, B harmonically, then N is the infinite point of the line AB. Exercise 1. If X, Y divide B, C harmonically, then B, C divide X, Y harmonically. 2. Given a point X on the line BC, construct its harmonic associate with respect to the segment BC. Distinguish between two cases when X divides BC internally and externally. 1 3. The centers A and B of two circles A(a) and B(b) are at a distance d apart. The line AB intersect the circles at A and B respectively, so that A, B are between A, B. 1Make use of the notion of centers of similitude of two circles. 1.1 Preliminaries 5 A B A B 4. Given two fixed points B and C and a positive constant k =1, the locus of the points P for which |BP| : |CP| = k is a circle. 1.1.5 Homothety Given a point T and a nonzero constant k, the similarity transformation h(T,k) which carries a point X to the point X on the line TX satisfying TX : TX = k :1is called the homothety with center T and ratio k. Explicitly, h(T,k)(P )=(1− k)T + kP. Any two circles are homothetic. Let P and Q be the internal and external centers of ( ) ( ) h( r ) h( − r ) similitude of two circles O R and I r . Both the homotheties Q, R and P, R transform the circle O(R) into I(r). 6 The Circumcircle and the Incircle 1.1.6 The power of a point with respect to a circle The power of a point P with respect to a circle C = O(R) is the quantity C(P ):=OP 2 − R2. This is positive, zero, or negative according as P is outside, on, or inside the circle C.Ifit is positive, it is the square of the length of a tangent from P to the circle. Theorem (Intersecting chords) If a line L through P intersects a circle C at two points X and Y , the product PX· PY (of signed lengths) is equal to the power of P with respect to the circle. T O X Y P T 1.2 Menelaus and Ceva theorems 7 1.2 Menelaus and Ceva theorems 1.2.1 Menelaus and Ceva Theorems Consider a triangle ABC with points X, Y , Z on the side lines BC, CA, AB respectively. Theorem 1.1 (Menelaus). The points X, Y , Z are collinear if and only if BX CY AZ · · = −1. XC YA ZB A Y Z X B C Theorem 1.2 (Ceva). The lines AX, BY , CZ are concurrent if and only if BX CY AZ · · =+1. XC YA ZB A Z Y P B X C 8 The Circumcircle and the Incircle 1.2.2 Desargues Theorem As a simple illustration of the use of the Menelaus and Ceva theorems, we prove the fol- lowing Desargues Theorem: Given three circles, the exsimilicenters of the three pairs of circles are collinear. Likewise, the three lines each joining the insimilicenter of a pair of circles to the center of the remaining circle are concurrent. X A Y C P Y Z X B Z We prove the second statement only. Given three circles A(r1), B(r2) and C(r3), the insimilicenters X of (B) and (C), Y of (C), (A), and Z of (A), (B) are the points which divide BC, CA, AB in the ratios BX r2 CY r3 AZ r1 = , = , = .
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