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Math Help - Geometry problem

  1. #1
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    Fast Help!

    I have to complete my homework in 2 days and this problem I found most difficult:

    Let ABC be a right triangle with hypotenuse BC. Suppose that M is the midpoint of BC and H is the feet of the perpendicular dropped from A onto BC. A point P, distinct from A, is chosen on the opposite ray of ray AM. Let the line through H perpendicular to AB intersect PB at Q; and let the line through H perpendicular to AC meet PC at R. Prove that A is the orthocenter of triangle PQR.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by thduong711
    I have to complete my homework in 2 days and this problem I found most difficult:

    Let ABC be a right triangle with hypotenuse BC. Suppose that M is the midpoint of BC and H is the feet of the perpendicular dropped from A onto BC. A point P, distinct from A, is chosen on the opposite ray of ray AM. Let the line through H perpendicular to AB intersect PB at Q; and let the line through H perpendicular to AC meet PC at R. Prove that A is the orthocenter of triangle PQR.
    See diagram. By definition of the Orthocentre to prove that A is the
    orthocentre of triangle PQR it is sufficient to prove that QA extended
    is a normal to PR, and that RA extended is a normal to PQ.

    Now it is sufficient to prove that in general QA extended is a normal PR,
    as by an equivalent argument RA will be a normal to PQ.

    Now the best way to proceed as far as I can see is to introduce
    coordinates, and use coordinate geometry to show that angle QSR
    in the diagram is a right angle.

    ==================================================

    <<You will need to check the algebra in this carefully>>

    Let A be the origin (0,0), B be (0,b) and C be (c,0).

    Then M is (c/2,b/2).

    The slope of CB is -b/c, so the slope of AH is c/b, in fact AH
    is the line y=(c/b)x. The line CB has equation y=(-b/c).x+b,
    so H is the point of intersection of:

    y=(c/b).x,

    and

    y=(-b/c).x+b.

    Which is the point (b^2.c/(b^2+c^2), b.c^2/(b^2+c^2)).

    Now P lies on AM, and may be written as -\mu.(c/2,b/2) for some
    \mu>0.

    To finish find the equation on the line through P and B, and from that
    find the coordinates of Q. Find the slope of the line through Q and A,
    which should be minus the slope of the line through P and C.


    RonL
    Attached Thumbnails Attached Thumbnails Geometry problem-orthocentre2.jpg  
    Last edited by CaptainBlack; December 20th 2005 at 10:22 AM.
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  3. #3
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    Sorry,but do you have another solution that just using similar/congruent triangles, circles...etc...I haven't learn about your way to solve this, so I don't understand that much, and I think my teacher has another more simple solution
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