If ABCD is a quadrilateral, then its midpoint quadrilateral MNOP is the quadrilateral with vertices at the midpoints of the sides: M = midpoint AB, N = midpoint BC, O = midpoint CD, P = midpoint DA..
For any quadrilateral ABCD, prove that the midpoint quadrilateral of ABCD is a parallelogram.
Draw a triangle ABC and construct midpoints D, E, F as in the figure. Let G be the intersection of the two medians CF and BD. Then let H be the midpoint of BG and I be the midpoint of CG.
a) Prove that DFHI is a parallelogram.
b) Then find the ratio BG/BD and also CG/CF.
c) Now repeat this construction for medians BD and AE, intersecting in J. Can you prove G = J? If so, this will prove that the medians are concurrent. Explain why.
Let ABCD be a rectangle, with midpoints M of AB and N of CD. Now suppose ABCD has the property that if you fold it in half along MN, i.e., the rectangle AMND is similar to the original one. Find the ratio of length to width of this rectangle.
Given angle AOB, |OA| = a, |OB| = b and AB = c, suppose that A' is a point on ray OA and B' is a point on ray OB.
a) Prove that triangle AOB is similar to triangle A'OB' if and only if there is a constant K > 0 so that OA' = Ka and OB' = Kb. In this case, what is the length of A'B'?
b) Prove that triangle AOB is similar to triangle B'OA' if and only if there is a constant K > 0 so that OA' = K/a and OB' = K/b. In this case, what is the length of A'B'?
c) Prove that the two triangles are not similar (for any ordering of the vertices) if neither (a) or (b) is true.\
