Separating Truth From Ficticious Geometry...Nein!
Your Task: Are these conjectures true or false? If true, can you prove them, using either algebra or geometry (or both!). If false, why?
Conjecture 1: The midpoint of the hypotenuse of a right triangle is equidistant from all three vertices.
Conjecture 2: The line joining the midpoints of nonparallel sides of a trapezoid is parallel to the third side and equal to half its length.
Conjecture 3: If D is the midpoint of side BC of triangle ABC, then AB^{2}+AC^{2} = 2(AD^{2}+DC^{2}).
Conjecture 4: The lines joining midpoints of opposite sides of a quadrilateral and the line joining the midpoints of the diagonals all bisect each other; hence the three lines are concurrent.
Conjecture 5: The sum of the squares of the sides of a parallelogram equals the sum of the squares of its diagonals.
Conjecture 6: Three times the sum of the squares of the sides of a triangle equals four times the sum of the squares of the medians.
Conjecture 7: If two medians of a triangle are congruent, the triangle is isosceles.
Conjecture 8: If the sum of the squares of the distances from any point of the plane to two opposite vertices of a parallelogram equals the sum of the squares of the distances from that point to the other two vertices, the figure is a rectangle.
Conjecture 9: If two opposite sides of a quadrilateral intersect at a point P and the other two pair of opposite sides intersect at a point Q, the midpoint of segment PQ is collinear with the midpoints of the quadrilateral's diagonals.
Source: Adapted from G. Merriman's To Discover Mathematics, 1942, pp. 388, 390
Hint: For each conjectures, a good place to begin is to explore it using Geometers SketchPad.
Key needs are to label all relevant points so that you can invoke both algebraic and geometric ideas.
Solution Commentary: No solutions provided....except to tell you that all of the conjectures are true...and can be proven using a combination of algebraic and geometric tools.
