1. Special Midpoint Quadrilaterals
(a) Consider this statement: If the midpoint quadrilateral of quadrilateral ABCD is a rectangle, then ABCD is a kite. Prove it if it is true. If it is false, give a counterexample and, if you can, explain for which quadrilaterals the midpoint quadrilateral is a rectangle.
(b) Consider this statement: If the midpoint quadrilateral of ABCD is a rhombus, then ABCD is a rectangle. Prove it if it is true. If it is false, give a counterexample and, if you can explain for which quadrilaterals the midpoint quadrilateral is a rhombus.
2. Circumcenter of an Isosceles Triangle
Given an isosceles triangle ABC, with |AB| = |AC| = b and |BC| = a, what is the radius R of the circumcircle of ABC? (The answer should be in terms of a and b.)
Suppose OAB is a triangle with |OA| = a, |OB| = b, and |AB| = k, let A' be the point on OA with |OA'| = 6/a and let B' be the point in OB with |OB'| = 6/b. Find the distance |A'B'| (this will be a formula involving explicit real numbers and a, b, and k, no other quantities). Show your reasoning.
4. Circumscribing circles given 3 equal segments and 2 equal angles (10 points)
In this problem and the next one, we will consider a figure made of 4 points A, B, C, D and 3 segments in which angle ABC = angle BCD and these 3 segments have equal length: |AB| = |BC| = |CD|. Two examples of such a figure are figure below.
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