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.


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