520. Given triangle ABC, let F be the point where segment BC meets the bisector of angle BAC, Draw the line through B that is parallel to segment AF, and let E be the point where this parallel meets the extension of segment CA. (a) Find the four congruent angles in your diagram. (b) How are the lengths EA, AC, BF, and FC related? (c) The Angle-Bisector Theorem: How are the lengths AB, AC, BF, and FC related? 520. Given...
Points W and X are chosen on the side AB of triangle ABC and points Y and Z are chosen on side AC. Suppose that cr(A,W,X,B)=cr(A,Y,Z,C) and that WY is parallel to XZ. Prove that XZ is parallel to BC. Hint: let T be the point where the parallel to XZ through B meets line AC. Note that neither a nor Y can lie on segment TC and use excercise 3C.2 to show that T is C. cr=cross ratio
Additional problem 1 Let AABC be a triangle, let be the bisector of the angle ZCAB Let P be the intersection of and BC. Let R be the point on the line AB such that AR-AC, and let X-APnRC. Let Q denote the intersection point between the line through B and X and AC. (a) Show that the triangle AARC is isosceles, and deduce that RX-XC. (b) Apply Menelaus's theorem to the triangle AARC with the line through B, X,...
hint for d): consider a point D such that M is the midpoint of CD. Which segments are congruent here? Do you see a triangle with all three side lengts given. Could you please write some instructions on the side so I know how to follow your solution? 5. Given a triangle ABC, let M be the midpoint of the segment AB. The segment CM is called the median of the triangle. Let T be the point on the line...
(1 point) Let T'be the region inside the triangle with vertices (0, 0), (3, 0) and (3, 2), and let f(x, y) be the function which is 0 outside of T and f(x, y)-38 SK y Ior (x, y) inside T. 3888 Then E(XY)- 4.8
2. Consider a triangle ABC. Let M denote the midpoint of side AC. If BM - AM, show that angle B is a right angle. (10 points)
23.4 Let ABC be any triangle, let DE be a line parallel to the base, and let F be any point on DE. Show that the area of the union of the two triangles DBF and ECF is less than or equal to one-fourth the area of the whole triangle, with equality if and only if D and E are the midpoints of AB and AC. 23.4 Let ABC be any triangle, let DE be a line parallel to the...
I need help doing a doing two column for these two propositions. Book 1 Proposition 7: Given two straight lines constructed from the ends of a straight line and meeting in a point, there cannot be constructed from the ends of the same straight line, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each equal to that from the same end. Book 3 Proposition 14:...
28 Consider (O:OAOB) an orthonormal system in space. Let G be the center of gravity of triangle ABC. 1° Calculate the coordinates of G 2°Consider the points A' (2 ;0:0) ,B, (0:2:0) and C" (0:0,3). a) Verify that these three points define a plane. b) Write a system of parametric equations of the plane (A'BC'). 3 Write a system of parametric equations of line (AC). 4° Verify that K (4:0-3) is the trace of the line (AC) with the plane...
Question B Diagram NOT accurately drawn 2 2b APB is a triangle. N is a point on AP. AB- a AN 2b NP-b (a) Find the vector PB, in terms of a and b. 3b 10 B is the midpoint of AC. M is the midpoint of PB. *(b) Show that NMC is a straight line. Diagram NOT accurately drawn 2 2b APB is a triangle. N is a point on AP. AB- a AN 2b NP-b (a) Find the...