Introduction Parabolas Ellipses Projectivity Geometric Approaches to Conics David Altizio CMU Undergraduate Summer Lecture Series 22 June 2018 B A Some properties of cyclic quadrilaterals: ◦ \ABC + \ADC = 180 , ◦ \BCD + \BAD = 180 \ABD = \ACD, etc. (Ptolemy) AB CD + AD BC = AC BD CD · · · The key is that these properties also go the other way - in other words, e.g. \ABD = \ACD implies ABCD is cyclic! Introduction Parabolas Ellipses Projectivity Geometry Review: Cyclic Quadrilaterals A quadrilateral ABCD is cyclic if there exists a circle passing through A, B, C, and D. B A CD The key is that these properties also go the other way - in other words, e.g. \ABD = \ACD implies ABCD is cyclic! Introduction Parabolas Ellipses Projectivity Geometry Review: Cyclic Quadrilaterals A quadrilateral ABCD is cyclic if there exists a circle passing through A, B, C, and D. Some properties of cyclic quadrilaterals: ◦ \ABC + \ADC = 180 , ◦ \BCD + \BAD = 180 \ABD = \ACD, etc. (Ptolemy) AB CD + AD BC = AC BD · · · B A CD Introduction Parabolas Ellipses Projectivity Geometry Review: Cyclic Quadrilaterals A quadrilateral ABCD is cyclic if there exists a circle passing through A, B, C, and D. Some properties of cyclic quadrilaterals: ◦ \ABC + \ADC = 180 , ◦ \BCD + \BAD = 180 \ABD = \ACD, etc. (Ptolemy) AB CD + AD BC = AC BD · · · The key is that these properties also go the other way - in other words, e.g. \ABD = \ACD implies ABCD is cyclic! Introduction Parabolas Ellipses Projectivity A Sample Applications Example Let ABC be a triangle with ABC = 90◦. Point D lies on AB, 4 \ and E is the foot of D onto AC. Then \ABE = \ACD. A E D BC Introduction Parabolas Ellipses Projectivity Two Sample Applications Example 1 Let ABC be a triangle with ABC = 90◦. Point D lies on AB, 4 \ and E is the foot of D onto AC. Then \ABE = \ACD. A E D BC Introduction Parabolas Ellipses Projectivity What is a parabola? A parabola is the locus of points X which have equal distances P to some fixed point F (called the focus of ) and some fixed line ` (called the directrix of ). P P X F ` P Introduction Parabolas Ellipses Projectivity Reflection Property for Parabolas Theorem Let m denote the tangent to at the point X . Then m bisects P \FXP. X F m ` P Introduction Parabolas Ellipses Projectivity Reflection Property for Parabolas Theorem Let m denote the tangent to at the point X . Then m bisects P \FXP. X F Y m ` Q P Introduction Parabolas Ellipses Projectivity Reflection Property for Parabolas Corollary The reflection of the focus F over the tangent line m lies on the directrix of . P X F m ` P Introduction Parabolas Ellipses Projectivity Simson Lines Theorem (Simson) Let ABC be a triangle with circumcircle !, and let P be a point in the plane. Then the projections of P onto the sides of ABC are 4 collinear iff P lies on !. Z A P Y ω BC X Introduction Parabolas Ellipses Projectivity Simson Lines Theorem Let H be the orthocenter of ABC. Then for any P on (ABC), the Simson line of P wrt ABC4 bisects PH. 4 Z A P Y H ω BC X Introduction Parabolas Ellipses Projectivity Applications of Simson Lines Consider the following scenario, where the pairwise intersection points of three distinct tangents to a parabola form a triangle ABC. We wish to find interesting relationshipsP between and ABC. P 4 P R F Q B C A Introduction Parabolas Ellipses Projectivity Applications of Simson Lines P R F B Q C ` A Introduction Parabolas Ellipses Projectivity Applications of Simson Lines Theorem The focus F of lies on the circumcircle of ABC. P 4 P R F Q B C A Introduction Parabolas Ellipses Projectivity Applications of Simson Lines Theorem The orthocenter H of ABC lies on the directrix of . 4 P P R F Q B H C ` A Introduction Parabolas Ellipses Projectivity What is an ellipse? An ellipse is the locus of points P in the plane such that the sum E of the distances from P to two fixed points F1 and F2 (called the foci of ) is a constant. E P F2 F1 E Introduction Parabolas Ellipses Projectivity Reflection Property for Ellipses Theorem Let ` denote the tangent to an ellipse at a point P. Then the E acute angles made by F1P and F2P with respect to ` are equal. P ` F2 F1 E Introduction Parabolas Ellipses Projectivity Reflection Property for Ellipses F10 P ` F2 F1 E Introduction Parabolas Ellipses Projectivity Isogonal Property of Ellipses Theorem Let XP and XQ be tangents to the ellipse . Then E \F1XP = \F2XQ. X P Q F2 F1 E Introduction Parabolas Ellipses Projectivity Isogonal Property for Ellipses X F10 P F20 Q F2 F1 E Introduction Parabolas Ellipses Projectivity Distances from Foci to Tangent Theorem Let be an ellipse with foci F1 and F2. Let ` be an arbitrary tangentE to at a point P. Then E 2 2 PF1+PF2 F1F2 dist(F1; `) dist(F2; `) = ; · 2 − 2 in particular, this quantity does not depend on P. P ` F2 F1 E Introduction Parabolas Ellipses Projectivity Distances from Foci to Tangent P2 P P1 X Q F2 F1 E Introduction Parabolas Ellipses Projectivity An Application (aka shameless advertising) CMIMC 2018 G9, Gunmay Handa Suppose 1 = 2 are two intersecting ellipses with a common focus E 6 E X ; let the common external tangents of 1 and 2 intersect at a E E point Y . Further suppose that X1 and X2 are the other foci of 1 E and 2, respectively, such that X1 2 and X2 1. If E 2 E 22 E X1X2 = 8; XX2 = 7, and XX1 = 9, what is XY ? X Y X1 X2 Introduction Parabolas Ellipses Projectivity An Application (aka shameless advertising) X Y X1 X2 2 If we work with the usual R plane, however, then parallel lines remain parallel because transformations are injective. Definition The line at infinity is a \line" which has points corresponding to intersections of parallel lines (one for each direction). The normal plane joined with the line at infinity is called the projective plane. For the remainder of this talk, we will be working in the projective plane. Introduction Parabolas Ellipses Projectivity Projective Transformations A projective transformation Φ is a transformation of the plane which preserves lines, i.e. A, B, and C are collinear iff Φ(A), Φ(B), Φ(C) are. Definition The line at infinity is a \line" which has points corresponding to intersections of parallel lines (one for each direction). The normal plane joined with the line at infinity is called the projective plane. For the remainder of this talk, we will be working in the projective plane. Introduction Parabolas Ellipses Projectivity Projective Transformations A projective transformation Φ is a transformation of the plane which preserves lines, i.e. A, B, and C are collinear iff Φ(A), Φ(B), 2 Φ(C) are. If we work with the usual R plane, however, then parallel lines remain parallel because transformations are injective. For the remainder of this talk, we will be working in the projective plane. Introduction Parabolas Ellipses Projectivity Projective Transformations A projective transformation Φ is a transformation of the plane which preserves lines, i.e. A, B, and C are collinear iff Φ(A), Φ(B), 2 Φ(C) are. If we work with the usual R plane, however, then parallel lines remain parallel because transformations are injective. Definition The line at infinity is a \line" which has points corresponding to intersections of parallel lines (one for each direction). The normal plane joined with the line at infinity is called the projective plane. Introduction Parabolas Ellipses Projectivity Projective Transformations A projective transformation Φ is a transformation of the plane which preserves lines, i.e. A, B, and C are collinear iff Φ(A), Φ(B), 2 Φ(C) are. If we work with the usual R plane, however, then parallel lines remain parallel because transformations are injective. Definition The line at infinity is a \line" which has points corresponding to intersections of parallel lines (one for each direction). The normal plane joined with the line at infinity is called the projective plane. For the remainder of this talk, we will be working in the projective plane. Introduction Parabolas Ellipses Projectivity Projective Transformations One can consider projective transformations as central projections between planes. It turns out that all projective transformations must be of this form. Given a circle ! and a point P, there exists a projective transformation sending ! to a circle !0 and sending P to the center of !0. Given a circle ! and a line `, there exists a projective transformation sending ! to a circle and sending ` to the line at infinity. Introduction Parabolas Ellipses Projectivity Facts about Projective Transformations For any quadruples of points A, B, C, D and A0, B0, C 0, D0 in general position, there exists a projective transformation sending A A0, B B0, C C 0, and D D0. ! ! ! ! Given a circle ! and a line `, there exists a projective transformation sending ! to a circle and sending ` to the line at infinity. Introduction Parabolas Ellipses Projectivity Facts about Projective Transformations For any quadruples of points A, B, C, D and A0, B0, C 0, D0 in general position, there exists a projective transformation sending A A0, B B0, C C 0, and D D0. ! ! ! ! Given a circle ! and a point P, there exists a projective transformation sending ! to a circle !0 and sending P to the center of !0. Introduction Parabolas Ellipses Projectivity Facts about Projective Transformations For any quadruples of points A, B, C, D and A0, B0, C 0, D0 in general position, there exists a projective transformation sending A A0, B B0, C C 0, and D D0.