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2015-02-19T16:34:08+05:30

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In right angled triangle ABE,
AB² = AE² + BE²
      = (AC+CE)² + BE²
      = AC² + CE² + 2.AC.CE + BE²
      = AC² + (CE²+BE²) + 2.AC.CE     (in ΔBCE, CE²+BE² = BC²)
⇒AB² = AC² + BC² + 2.AC.CE --------------------(1)

Similarly in right angled triangle ADB,
AB² = AD² + BD²
      = AD² + (BC+CD)²
      = AD² + BC² + 2.BC.CD + CD²
      = (AD² + CD²) + BC² + 2.BC.CD    (In ΔACD, AD²+CD² = AC²)
⇒AB² = AC² + BC² + 2.BC.CD --------------------(2)

From equations (1) and (2);
AC² + BC² + 2.AC.AE = AC² + BC² + 2.BC.CD
⇒ AC.CE = BC.CD   ----------------------------------(3)

From triangle ADB,
AB² = AD² + BD²
      = (AC² - CD²) + (BC+CD)²    (In ΔACD, AC² = AD²+CD²)
      = AC² - CD² + BC² + CD² + 2.BC.CD
      = AC² + BC² + 2.BC.CD
      = AC² + BC² + BC.CD + BC.CD      (from equation 3)
      = AC² + BC² + BC.CD + AC.CE
      = AC² + AC.CE + BC² + BC.CD
      = AC(AC+CE) + BC(BC+CD)
AB² = AC.AE + BC.BD    (PROVED)
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