Geometry challenge

[krypt]

Hello, I'm new here (to posting not to the forums).
The following is a challange proposed by our math teacher.
The work is due tomorrow morning.

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ABC is a right triangle. (BAC=90°)
[AH] is an "altitude" (AH_|_BC)
I is the center of [BC]
J is the center of [AH]
Starting from A, lead a perpendicular to (IJ) which intersects (BC). The intersection is F.
(JF) intersects (AB), the intersection is E.
Prove that (HE) is perpendicular to (AB).
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I have tried working in the FHXA rectangle (X being H's orthogonal projection on (AC), my intuition keeps taking me there but I'm still having no luck solving it.

Thank you for your help,
I do appreaciate it.

I am sorry for the ill terms I might have used as I am French educated.

 

[krypt]

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[stapel]

Note to tutors: For those of you whose browsers don't like the funky FTP-but-not-FTP URL, here's a description of the graphic:

Draw a right triangle, ABC, with AC being the base (A to the left, C to the right), AB being the height, and the right angle being angle BAC.

From A, erect the perpendicular to BC, meeting BC at H, so AHC is another right triangle, and AHC shares the angle at C. The base is HC, the altitude is AH, the hypotenuse is AC, and the right angle is angle AHC.

Let I be the midpoint of BC and let J be the midpoint of AH. Contruct the line through I and J.

From A, contruct the perpendicular to line IJ, extending the line until it meets BC (or the extension of BC). Label this intersection point as F, and construct the segment JF. Label the point where JF intersects AB as point E.

Prove that line HE is perpendicular to line AB.

Note: In the above, I assumed that "center" meant "midpoint". ("Center" is usually used in the context of area, such as in circles, while "midpoints" are, I believe, always the points marking the middle of line segments.)

The first bit of the construction looks something like this:

  F
  * .
   '  * . B
    '     * .   H
     '  E |   * .   I
      '   |  '    * . 
       '  | 'J        * .
        ' |'              * .
          A-------------------C