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.
+128473
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
