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Re: problem from Komal

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  • vprasad_nalluri
    ... Show ... Let BH,IF intersect at X; CH,IE intersect at Y The figure IXHY enclosed by parallels BH,IE and CH,IF is a parallelogram So HI bisects XY...(i)
    Message 1 of 2 , May 2, 2004
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      --- In Hyacinthos@yahoogroups.com, "ben_goss_ro" <ben_goss_ro@y...>
      wrote:
      > Here's a nice problem:
      >
      > Let the incircle of ABC touch CA and AB at E and H respectively.
      Show
      > that if H is on EF then HI passes through the midpoint of BC. (H is
      > the orthocenter and I is the incenter)

      Let BH,IF intersect at X; CH,IE intersect at Y
      The figure IXHY enclosed by parallels BH,IE and CH,IF is a
      parallelogram
      So HI bisects XY...(i)
      Angles IFE,IEF are equal, so are corresponding angles EHY,FHX
      So triangles XHF, YHE are similar,
      HX/HY = FH/EH = Sin FAH / Sin EAH = Cos B / CosC = HB / HC
      Follows XY is parallel to BC ...(ii)
      Hence from (i) & (ii)
      HI bisects BC

      Vijay
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