Sunday, July 19, 2009

Problem 325. Isosceles triangle, Altitude, Incircle, Excircle

Proposed Problem
Click the figure below to see the complete problem 325 about Isosceles triangle, Altitude, Cevian, Incircle, Excircle with equal radius.

Geometry Problem 325. Isosceles triangle, Altitude, Cevian, Incircle, Excircle.
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Complete Problem 325
Level: High School, SAT Prep, College geometry

Posted by Antonio Gutierrez at 2:22 PM
Labels: , , , , , ,

4 comments:

  1. c .t . e. oJanuary 18, 2010 at 2:23 PM

    let be M midpoint of AC, K point BC meet circle F

    draw MT tangent to circle F =>
    T, F, K are at the same line (FK perpend to BC, & FT to MT) => TK diameter ( MT//BK)

    extend TM to G on AH => MG middle line of tr ACH
    => G midpoint of AH =>

    GH = AH = TK = 2r => AH = 4r

    ReplyDelete
  2. AnonymousMarch 2, 2010 at 12:05 AM

    Why "T, F, K are at the same line (FK perpend to BC, & FT to MT)"???

    ReplyDelete
  3. c .t . e. oMarch 11, 2010 at 1:13 PM

    from a point , F, we can draw one ( only ) perpendicular to two parallel lines ( GT//HK )

    ReplyDelete
  4. c .t . e. oMarch 20, 2010 at 1:04 PM

    draw ST tg to F and // to AC
    name G, K, L, E meet AD, AB, BD
    name M, N, R, P, F meet DC, DS, ST, CT
    ------------------------------------------
    ▲AEG=▲AEK=▲RFT=▲PFT => AK=AG=RT=PT ( anlge & radius )
    ▲GED=▲DEL=▲DMF=▲DNF => GD=DM=DL=DN
    MC = CP tg from C, NS = SR tg from S
    ------------------------------------------
    BN = BP => BD + DM = BC + MC =>
    BD + DM = AB + MC (1)
    ------------------------------------------
    SADB + SBST = SABC + SDCTS
    (AB+AD+BD)∙r+(BS+ST+BT)∙r=BC∙h+(DC+CT+ST+DS)∙r
    (AB+AD+BD+BS+BT)∙r=BC∙h+(DC+CT+BT)∙r
    (AB+AD+BD+BD+DM+NS+BC+CT)∙r=BC∙h+(DC+CT+BT)∙r
    (3∙AB+AD+BD+MC+NS)∙r=BC∙h+(DC+DS)∙r
    (3∙AB+AG+DM+BK+DN+MC+NS)∙r=BC∙h+(DC+DS)∙r
    (4∙AB+DC+DS)∙r=BC∙h+(DC+DS)∙r
    4∙AB∙r = BC∙h

    4∙r = h
    --------------------------------------------

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