Wednesday, November 28, 2012

Fox 354


Roland Sampy has submitted this problem.
It looks right, but not confirmed yet.

7 comments:

  1. Two equilateral triangles OPQ and ORS of respective sides a and b.

    Case 1: a=b
    then OP=OQ=OR=OS=a=b=r, i.e. (P,Q,R,S) lie on the same circle, irrespective of triangles' orientation.
    Hence a^2+a.b+b^2=r^2+r.r+r^2=3.r^2

    Case 2: a<>b
    Assume PQ and RS are not parallel. (P,Q,R,S) on the same circle implies that circle's center I is the intersection of PQ and RS bisectors, which is O by construction. Since OP=a<>b=OR we have a contradiction. Therefore PQ//RS.

    PQ and RS share a common bisector as depicted on the following figure:
    http://bleaug.free.fr/8foxes/8foxes354.png

    Since P,J,S belong to circle centered in I we have QPR=QJR=QSR=QIR/2=60°. Hence QR=r.sqrt(3)

    PQ//RS implies QOR=120°, therefore in triangle OQR:
    QR^2=OQ^2+OR^2-2.OQ.OR.cos(QOR)
    which can be rewritten as:
    3.r^2=a^2+b^2+a.b

    bleaug

    ReplyDelete
  2. to be complete, we should also consider the (sub-)case where PQ and RS lie on the same side w.r.t. O as depicted in:
    http://bleaug.free.fr/8foxes/8foxes354b.png

    If a and b represent positive length values, formula becomes:
    3.r^2=a^2+b^2-a.b

    If we adopt the convention that (a.b) = scalar product (OQ.OS) including sign, the above formula holds in all cases.

    bleaug

    ReplyDelete
  3. Two equilateral triangles OPQ and ORS of respective sides a and b.
    then arc(PQ) + arc(RS) = 2 . 60 degree =120 degree
    but arc(PR) = arc(QS) .
    Then arc(PR) = arc(QS) = 120 degree.
    and PR = QS = r.sqrt(3) .
    In the triangle OPR angle(POR) = 120 degree.
    Then OP^2 + OR^2 + OP.OR = PR^2
    and a^2 + b^2 + a.b = r.sqrt(3) όπερ έδει δείξαι.
    Philip Pydnaios , from Hellas

    ReplyDelete
  4. Tantrik Astrologer Ramdas Ji. in India Best Vashikaran Specialist, Contact Me : +91-9587008635 If you are looking for the cure of your problems with help of vashikaran and black magic then vashikaran specialist ramdas baba ji can be the best motivator in all sorts. You just have to call us and he can perform all the rituals to solve your problem. The spell and mantras of their vashikaran and black magic is too powerful to make your dream come true.

    Love Problem Solution
    Enemy Kill Mantra
    Kala Jadu
    Black Magic for Destroy Enemy
    Black Magic Removal Specialist
    Aghori Mantra to Kill Enemy
    Boyfriend Vashikaran Specialist
    Enemy Died Mantra
    Black Magic to Kill Enemy
    Enemy Died Mantra by Black Magic
    Destroy Enemy By Maran Mantra
    How Can i Kill or Destroy My Enemy
    How To Get Your Ex Boyfriend Back
    Vashikaran Specialist Astrologer
    Maran Mantra to Kill Enemy
    Black Magic To Kill A Person
    Husband Wife Relationship Problem Solution

    ReplyDelete
  5. Tantrik Astrologer Ramdas Ji. in India Best Vashikaran Specialist, Contact Me : +91-9587008635 If you are looking for the cure of your problems with help of vashikaran and black magic then vashikaran specialist ramdas baba ji can be the best motivator in all sorts. You just have to call us and he can perform all the rituals to solve your problem. The spell and mantras of their vashikaran and black magic is too powerful to make your dream come true.

    Childless Problem Solution
    Kala Jadu Specialist Aghori Baba
    Famous Astrologer Tantrik
    Kamakhya Sindoor By Vashikaran Mantra
    Inter Caste Love Marriage Solution
    Get Your Love Back
    Black Magic Specialist
    Lottery Number Specialist
    Black Magic to Get Lost Love Back
    Get Lost Love Back By Astrology
    Black Magic Mantra To Spells Destroy Enemy Kill
    Husband Wife Problem Solution
    Love Vashikaran Specialist
    Love Marriage Specialist
    Get Love Back By Vashikaran
    Divorce Problem Solution
    Kamdev Vashikaran Mantra

    ReplyDelete