Monday, March 30, 2015

Problem 1104: Right Triangle, Incircle, Circumcircle, Inscribed Circle, Radius, Tangent

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Click the diagram below to enlarge it.

Online Math: Geometry Problem 1104: Right Triangle, Incircle, Circumcircle, Inscribed Circle, Radius, Tangent.

4 comments:

  1. HBMO1 is a rectangle.
    HD = EM, OD = OE imply ΔHDO and ΔMEO are congruent.
    So <DHO = <EMO.
    Also <BHM = <BMH = ½ HO1M = 45 deg.
    Follows HBMO1 is a square.
    So <DOH = <EOM = 45 deg and H, O, M are collinear.
    Implies HBMO1 is a square and O is its center.
    Hence x = O1M = 2 OE =2r = b + c – a.

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    Replies
    1. Referring to lines " Also <BHM = <BMH = ½ HO1M = 45 deg.
      Follows HBMO1 is a square.
      So <DOH = <EOM = 45 deg and H, O, M are collinear."
      Note that <DOH = <EOM = 45 deg if H, O, M are collinnear . Please provide explanation.
      Note that the solution never use given data " circle O, tangent to circle Q" . Please check it.

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  2. http://s25.postimg.org/59a5jt4qn/pro_1104.png
    This is a special case of general Sangaku problem and
    the general solution is
    x= r+ 2.d.(s-a).(s-c)/(b.s)…………..(1)
    see sketch below for detail
    http://s25.postimg.org/vvmm8s8xr/General_Sangaku_problem.png
    In our case we have d= ½. b.
    replace it in (1) and simplify we get
    x= r+ (s-a).(s-c)/s ………(2)
    in right triangle ABC (s-a)= ½(-a+b+c) and (s-c)= ½(a+b-c)
    so (s-a).(s-c)= ¼.(b^2- (a-c)^2)
    replace b^2= a^2+c^2 in above and simplify
    we get (s-a).(s-c)= ½. ac= area of triangle ABC= s.r
    so (s-a)(s-c)/s= r
    and x= 2.r= a+c-b

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  3. Draw OL // BC where L is on O1M

    Apply Pythagoras to right Tr. O1OL

    (b/2 - x)^2 = (x-c/2)^2 + (a/2-x)^2 from which we have upon simplification using b^2=c^2 + a^2,

    x = a+c-b which is anyway = to 2r since BDOE is a square of side r and AF = c-r and FC = a-r

    Sumith Peiris
    Moratuwa
    Sri Lanka

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