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# Geometry Proofs

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1) Show that the circumradius of a right triangle is equal to half the hypotenuse.

2) Show that the median to the hypotenuse of a right triangle is equal to half the hypotenuse.

3) Show that if a median of a triangle is one-half the side to which it is draw, then the triangle must be right.

4) Show that angle bisectors, perpendicular bisectors, and the medians all intersect at one point and only one point.

Nov 20, 2017

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1) Show that the circumradius of a right triangle is equal to half the hypotenuse.

Let   ABC  be a right triangle  and let B be the right angle

Let  B  = (0,0), A = (0, a)  and C =  (c ,0)

So......the midpoint of the hypotenuse is given by  (c/2, a/2)

And the distance ffrom this point to A  is    √ [ c^2 /4   + a^2/4]  = √ [ a^2 + c^2] / 2

And the distance from this point to C  is  √ [ c^2/4 + + a^2/4 ] =  √ [ a^2 + c^2] / 2

And the distance from this point to B  is √ [ a^2 + c^2] / 2

So.....a circle centered at the midpoint of the hypotenuse will pass through all three of the vertices since they are all equally distant from this midpoint   Nov 20, 2017