The Rectangle Proof
Nicole Mosteller EMAT 6690

This proof requires the application of properties of parallelograms and rectangles.

Given a right triangle XYZ with the median of the hypotenuse labeled M.

Figure 1: Given.

To complete this proof it is necessary to add a few lines.

Below, segment AY has been constructed parallel to segement ZX, and

segment XA has been constructed paralled to segment ZY.

Figure 2: The rectangle.

By definition, quadrilateral XAYZ is a parallelogram with right angles (because angle XZY is right).

XAYZ is better known as a rectangle.

The remainder of our proof involves the established properties of parallelograms and rectangles.

The figure below shows the rectangle with diagonals XY and AZ drawn.

Because XAYZ is a parallelogram, the diagonals bisect each other.

XM = YM and AM = ZM.

Because XAYZ is a rectangle, the diagonals are congruent.

XM = YM = AM = ZM.

Figure 3: XM = YM = AM = ZM.

By definition, point M is equidistant from points Z, X, and Y.

Figure 4: XM = YM = ZM.

So, the midpoint M is eqidistant from all of the vertices of the right triangle.



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