Gina wrote the following paragraph to prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side.
Given: ∆ABC
Prove: The midsegment between sides Line segment AB and Line segment BC is parallel to side Line segment AC.
Draw ∆ABC on the coordinate plane with point A at the origin (0, 0). Let point B have the ordered pair (x1, y1) and locate point C on the x-axis at (x2, 0). Construct point D so it is the midpoint of Line segment AB. Point D has coordinates at Ordered pair the quantity 0 plus x sub 1, divided by 2; the quantity 0 plus y sub 1, divided by 2 by the Midpoint Formula. Construct point E so it is the midpoint of Line segment BC. The ordered pair of point E is Ordered pair the quantity x sub 1 plus x sub 2, divided by 2; the quantity 0 plus y sub 1, divided by 2 by the Midpoint Formula. The slope of Line segment DE is found to be 0 through the application of the slope formula: The difference of y sub 2 and y sub 1, divided by the difference of x sub 2 and x sub 1 is equal to the difference of the quantity 0 plus y sub 1, divided by 2, and the quantity 0 plus y sub 1, divided by 2, divided by the difference of the quantity x sub 1 plus x sub 2, divided by 2 and the quantity 0 plus x sub 1, divided by 2 is equal to 0 divided by the quantity x sub 2 divided by 2 is equal to 0 When the slope formula is applied to Line segment AC the difference between y sub 2 and y sub 1, divided by the difference of x sub 2 and x sub 1 is equal to the difference of 0 and 0, divided by the difference of x sub 2 and 0 is equal to 0 divided by x sub 2 is equal to 0, its slope is also 0. Since the slope of Line segment DE and Line segment AC are identical, Line segment DE and Line segment AC are parallel by the Parallel Postulate.
What is the flaw in Gina’s proof?
Gina cannot construct the midpoint of a segment.
Segments DE and AC are parallel by definition of parallel lines.
The coordinates of D and E were found using the Distance between Two Points Postulate
The slope of segments DE and AC is not 0.

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The last one. The slope of DE is NOT 0

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