An Experiment in Reasoning Let ΔABC be given, and construct lines through the vertices par...

An Experiment in Reasoning Let ΔABC be given, and construct lines through the vertices parallel to the opposite sides, as shown in the figure. If P, Q, and R are the resulting points of intersection of these lines, several parallelograms are formed, one of them shown as a shaded region. (Recall that the opposite sides of a parallelogram are congruent.) Attempt to answer all the questions below, then see if your conclusion proves an important theorem in geometry.

(1) Is quadrilateral QABC a parallelogram? Is QA = CB?

(2) Is quadrilateral ACBR a parallelogram, and is CB = AR?

(3) Does it follow that A is the midpoint of segment ?

(4) What reasoning tells you whether B and C might be the midpoints of segments

(5) Would the perpendiculars to segments have to meet at some common point? What is that point, relative to ΔPQR?

(6) Recall that if a line is perpendicular to one of two parallel lines, it is also perpendicular to the other. Does the point of concurrency of Step 5 have any bearing on an important theorem about ΔABC?