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I have the following question:

 

Let a, b, and c be nonnegative real numbers, and let A=(0,0), B=(a,b), and C=(c,0) be points in the coordinate plane, such that AB=AC. Let D be the midpoint of ¯BC, let E be the foot of the altitude from D to ¯AC, and let F be the midpoint of ¯DE.

 

(a) Express the coordinates of points E and F in terms of a, b, and c.

(b) Show that line segments ¯AF and ¯BE are perpendicular.

 

And I am stuck on #b. I am almost done, but not sure how to prove that a^2+b^2=c^2. (not to be mistaken for the points)

 Nov 26, 2020
 #1
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(a) E = (a + c,0) and F = (a + c,b/2).

 

(b) The slope of AF is -a/b and the slope of BE is b/a.  The product of the slopes is -1, so they are perpendicular.

 Dec 12, 2020

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