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 Big B's theorem (Posted on 2012-10-14)
Brahmagupta*'s theorem states that if a cyclic quadrilateral ABCD has perpendicular diagonals, then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.

Today a high school student can prove it.

*Indian mathematician of 7th century

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 Solution Comment 2 of 2 |
`Let E be the intersection of the diagonalsAC and BD. Let F be the point on side CDsuch that line EF is perpendicular to sideCD. Let G be the intersection of line EFand side AB. We need to prove that   |GA| = |GB|.  <GAE = <BAC     -- Same angle       = <BDC     -- ABCD is concylic       = <EDF     -- Same angle       = <CEF     -- Similar rt. triangles       = <GEA     -- Vertical anglesTherefore, |GA| = |GE|.  <GBE = <ABD     -- Same angle       = <ACD     -- ABCD is concylic       = <ECF     -- Same angle       = <DEF     -- Similar rt. triangles       = <GEB     -- Vertical anglesTherefore, |GB| = |GE|.Hence, |GA| = |GB|.QED`

 Posted by Bractals on 2012-10-14 19:04:43

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