Sunday, August 3, 2008

Elearn Geometry Problem 154



See complete Problem 154
Triangle, Inradius, Circumradius, Chord. Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

5 comments:

  1. 1)in triangle DIC
    ang(DIC)=ang(BDC)=ang(BAC)=A;inscribed angles with endpoints B and C
    the lines BI and CI are bissectors ang(DIC)=(B+C)/2;ang(ICD)=A+B+C-A-(B+C)/2=(B+C)/2
    triangle DIC is isoscele DI=DC
    2)with the law of sines
    in triangle IBC: BC sin(C/2)=IB sin((B+C)/2)
    in triangle IDC: IC sin((B+C)/2)=ID sinA
    in triangle ABC: BC= 2RsinA and r= ICsin(C/2)
    therefore 2Rr=IB.ID

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  2. other questions related to this configuration
    a) prove that D is the circumcenter of triangle AIC
    b)lines AI,CI intersect the circumcercle of ABC at E and F.
    Prove that I is the orthocenter of DEF

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  3. (1) Since BD bisects angle B, D is the midpoint of arc ADC. Therefore B/2 = CAD = ACD, and ICD = C/2 + ACD = C/2 + B/2. On the other hand, DIC = IBC + BCI = B/2 + C/2. Hence DIC is isosceles and DI = CD.

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  4. http://ahmetelmas.wordpress.com/2010/05/15/geo-geo/

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  5. http://img208.imageshack.us/img208/1433/respuesta154.png

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