Exercises for Unit I I I Basic Euclidean concepts and theorems Default assumption 2 3 All points etc are assumed to lie in R or R I I I 1 Perpendicular lines and planes Supplementary background readings Ryan pp 16 18 Exercises to work 1 Suppose that P Q and T are three distinct planes and suppose that they have at least one point in common but do not have a line in common Prove that they have exactly one point in common 2 Suppose P and Q are two planes which intersect in the line L x U where the 1 dimensional vector subspace U spanned by the unit vector u Express these planes as translates of two dimensional subspaces with P x V and Q x W Let a and b be unit vectors in V and W respectively such that a and b are perpendicular or normal to u Prove that the cosine of the angle x a x x b is equal to the cosine of the angle between the normals to P and Q note that these normals are given by a u and b u Hint Express the dot product of the normals in terms of the dot product of a and b Apply the formula for v w y z derived in Section I 2 Note If we let P x a denotes the union of L with the set of all points on the same side of P as x a and we let Q x b denotes the union of L with the set of all points on the same side of Q as x b then the union of P x a and Q x b is an example of a dihedral angle and the result of the exercise states that two standard methods for defining the measure of this dihedral angle yield the same value 3 Let X be a point in the plane P Prove that there is a pair of perpendicular lines L and M in P which meet at X and that there is no line N in P through X which is perpendicular to both L and M Hint Try using linear algebra 4 Assume the setting of the previous exercise but also assume that P is contained 3 in R Prove that there is a unique line K through X which is perpendicular to both L and M 5 Ryan Theorem 19 pp 18 19 Let L and M be lines which intersect at Y and for each X in L Y let MX denote the foot of the unique perpendicular from X to M Prove that for each positive real number a there are exactly two choices of X for which d X MX a Hint Parametrize the line in the form Y t V for some nonzero vector V let W be a nonzero vector such that L and M lie in the plane determined by Y Y V and Y W with W perpendicular to V and express d X MX in terms of t and the length of W I I I 2 Basic theorems on triangles Supplementary background readings Ryan pp 60 67 Exercises to work Review of topics from Section II 4 Suppose that we are given ABC and DEF and let G and H denote the midpoints of BC and EF respectively Prove that 1 ABC 2 DEF if and only if that GAB HDE Suppose that ABC is an isosceles triangle with d A B d A C and D is a point of BC such that AD bisects BAC Prove that D is the midpoint of BC and that ADB ADC 90 3 Suppose we are given isosceles PRL with d R P d L P Let S and T be points on RL such that R S T d R S d L T and d P S d P T Prove that RTP LSP and PSR PTL 4 Suppose we are given two lines AE and CD and suppose that they meet at a point B which is the midpoint of AE and CD Prove that AC DE 5 Suppose that we are given lines AE BD and FG which contain a common point C and also satisfy A F B B C D and D G E Suppose also that d A C d E C and d B C d C D Prove that ABC EDC and AFC EGC Hint Part of the proof is to show that the betweenness properties A C E and F C G suggested by the drawing are true 6 Suppose that ABC is an isosceles triangle with d A B d A C and let D and E be points of AB and AC respectively such that d A D d A E Prove that BC DE 7 Suppose that we are given ABC and let D be a point in the interior of ABC such that AD bisects CAB BD bisects CBA and ADB 130 Find the value of ACB 8 Suppose that we are given points A B C such that A B C and let DE AC such that D B E CE AC and DE AD Prove that DAB BEC 9 Ryan Exercise 45 p 70 Prove the following result due to Heron of Alexandria Let P be a plane let L be a line let A and B be points on the same side of L in P and let C be the mirror image of B with respect to L formally choose C such that L is the perpendicular bisector of BC Define a positive real valued function f on L by f X d A X d X B Then the minimum value of f X occurs when X lies on AC Hint Why is d X B d X C and how is this relevant to the problem 10 Given ABC let X Y and Z be points on the open segments AB BC and AC respectively Prove that the sum of the lengths of the sides of ABC is greater than the sum of the lengths of the sides of XYZ 11 Given ABC let D and E be the midpoints of BC and AC respectively Prove that d D E d A B Given ABC let D be the midpoint of BC Prove that d A D d A B d A C Hint Let F be the midpoint of AB and apply the previous exercise 12 13 Given ABC let X be a point in the interior of ABC Prove that AXB BXC CXA 360 Hint There is one large triangle in the picture and it is split into three smaller ones the angle sum for each triangle is equal to 180 14 Prove the Sloping Ladder Theorem Suppose we are given right triangles ABC and DEF with right angles at C and F respectively such that the hypotenuses satisfy d A B d D E If d E F d B C then d A C d D F 15 In ABC one has d A C d B C If E is …