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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