In the question ABC is an obtuse triangle with C>
90 degree. AD is perpendicular to BC produced and BE is perpendicular to AC
produced.
Now take the triangle ADC: AD^2 = AC^2 -
CD^2
In triangle ABD: AD^2 = AB^2 -
DB^2
=> AC^2 - CD^2 = AB^2 -
DB^2
=> AB^2 = AC^2 + DB^2 - CD^2
...(1)
Now take the triangle BCE: BE^2 = BC^2 -
CE^2
In triangle BAE: BE^2 = AB^2 -
AE^2
=> BC^2 - CE^2 = AB^2 -
AE^2
=> AB^2 = BC^2 + AE^2 - CE^2
...(2)
Adding (1) and
(2)
2AB^2 = AC^2 + DB^2 - CD^2 + BC^2 + AE^2 -
CE^2
=> 2AB^2 = AC^2 + DB^2 - CD^2 + BC^2 + AE^2 -
CE^2
=> 2AB^2 = AC^2 + DB^2 - (DB - CB)^2 + BC^2 +
AE^2 - (AE - AC)^2
=> 2AB^2 = AC^2 + DB^2 - (DB^2
+CB^2 - 2*CB*DB)+ BC^2 + AE^2 - (AE^2 +AC^2-
2*AE*AC)
=>2AB^2 = AC^2 + DB^2 - DB^2 -CB^2 +
2*CB*DB+ BC^2 + AE^2 - AE^2 -AC^2+ 2*AE*AC
=> 2AB^2
= AC^2 -AC^2+ DB^2 - DB^2 -CB^2 + BC^2 + 2*CB*DB + AE^2 - AE^2 +
2*AE*AC
=> 2AB^2 = 2*CB*DB +
2*AE*AC
=> AB^2 = BD* BC + AC*
AE
Therefore we have the required result:
AB^2 = BD* BC + AC*
AE
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