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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 $\overline{BC},$ let $E$ be the foot of the altitude from $D$ to $\overline{AC},$ and let $F$ be the midpoint of $\overline{DE}.$

 

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

(b) Show that line segments $\overline{AF}$ and $\overline{BE}$ 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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