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# Geometry proof.

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Let A=(0,0), B=(a,b), and C=(c,0) be points in the coordinate plane for some numbers a,b, and c such that AB=AC. Let D be the midpoint of BC, E be the foot of the perpendicular from D to CA, and F be the midpoint of DE.

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

(b) Show that AF and BE are perpendicular.

Feb 26, 2020

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If  AB  =  AC

Then  the coordinates of of  C  must  be    ( sqrt  ( a^2 + b^2)  , 0) = ( c , 0)

But this implies  that  a^2 + b^2  = c^2

Which implies that   a^2  - c^2  =  -b^2

Which also  means that  (a - c) (a + c)  = - b^2

D  =  [ ( a + c)/2 , b/2 ]

E  =  [( a + c) /2, 0]

F =   [ (a+ c) / 2 , b/4 ]

Slope  of   AF  =   b/4                       b

_________   =   _______

(a + c)/2             2 (a + c)

Slope  of  BE    =           b                               b                         b                    2b

_______________ =     ___________  =  _______   =   ____

( a  -  (a+ c)  / 2)            a - a/2 - c/2         a/2 - c/2           a - c

Multiplying  top.bottom  of the last  fraction  by  a + c   we  have that

2b ( a + c)                   2b (a + c)                2 ( a + c)          -  2 ( a + c)

____________  =   ____________  =      _________  =   __________

(a - c) (a + c)              -  b^2                            -  b                    b

So  slope  of  AF  *  slope of  BE  =

b                          - 2 (a + c)

_________        *       _________   =     -  1

2(a + c)                         b

Which  means  that  AF  and BE  are perpendicular   Feb 26, 2020
edited by CPhill  Feb 26, 2020