MathDB
2018 BAMO D/2 parallel lines, tangents, P_7 coincides with P_1

Source:

August 26, 2019
geometryTangentsparallel

Problem Statement

Let points P1,P2,P3P_1, P_2, P_3, and P4P_4 be arranged around a circle in that order. (One possible example is drawn in Diagram 1.) Next draw a line through P4P_4 parallel to P1P2P_1P_2, intersecting the circle again at P5P_5. (If the line happens to be tangent to the circle, we simply take P5=P4P_5 =P_4, as in Diagram 2. In other words, we consider the second intersection to be the point of tangency again.) Repeat this process twice more, drawing a line through P5P_5 parallel to P2P3P_2P_3, intersecting the circle again at P6P_6, and finally drawing a line through P6P_6 parallel to P3P4P_3P_4, intersecting the circle again at P7P_7. Prove that P7P_7 is the same point as P1P_1. https://cdn.artofproblemsolving.com/attachments/5/7/fa8c1b88f78c09c3afad2c33b50c2be4635a73.png
Back to ProblemsView on AoPS