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How do I write a proof of this?
1: prove that the line segments joining the midpoints of successive sides of any quadrilateral form a parallelogram.
#2: prove that the line segments joining the midpoints of successive sides of any rectangle form a rhombus.
I don't necessarily need the entire proof, I just need to know how I can put these two things into a proof format.
Thanks!
1 Answer
- No MythologyLv 78 years ago
There are various approaches. If you are interested in Analytic Geometry, you could put an arbitrary (convex) quadrilateral in the plane with one vertex at the origin, another on the x-axis at (a, 0), and the remaining at some points (b, c) and (d, e). Then work out the length and slopes of the segments.
Another approach would be more along the lines of Synthetic Geometry. If you consider a (again, convex) quadrilateral with vertices ABCD, you can form the diagonal AC. Now triangles ABC and ADC would have a common base AC. You can use the fact that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. So the segment connecting the midpoints of AB and BC is parallel to AC and half its length. Similarly, the segment connecting midpoints of AD and DC is parallel to AC and half its length. These two segments are opposite sides of the new quadrilateral---and they are parallel and equal in length.
Play this same game with triangles DAB and DCB obtained by putting in the diagonal BD to ABCD.
This latter method gives you #2 for free since diagonals of a rectangle are equal.
