The Circle Proof
**Nicole Mosteller**
**EMAT 6690**

**This proof requires application
of the definitions in relation to the parts of a circle and properties
of inscribed angles.**

Given a right triangle XYZ with the
median of the hypotenuse labeled M.
__Figure 1:__ Given.

The first step in our proof is to inscribe
the right angle XZY in a circle.
The circle has the center to be point
M and the radius segment ZM.
__Figure 2:__ ZM is a radius of Circle M.

Remember the angle that we inscribed
is a right angle. Inscribed right angles intercept
arcs that measure 180 degrees (semicircles),
and this makes the hypotenuse of any inscribed right
triangle the diameter of the circle.
__Figure 3:__ XY is a diameter of Circle M.
Since XY is a diameter of Circle M,
and point M is the midpoint of XY,
by definition, XM and YM are radii
of Circle M.

Since segments ZM, XM, and YM are all
radii of the same circle, then
ZM = XM = YM.
__Figure 4:__ ZM = XM = YM.
By definition, point M is equidistant
from points Z, X, and Y.
So, the midpoint M is eqidistant from
all of the vertices of the right triangle.

Return
to Intro | Proof
#2 | Proof #3
| Proof #4 | Proof
#5 | To Nicole's Page