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In any triangle ABC,if a²,b²,c² be in AP.,then prove that cotA,cotB,cotC are also in AP

In any triangle ABC,if a²,b²,c² be in AP.,then prove that cotA,cotB,cotC are also in AP

Grade:11

1 Answers

Jayanth Kasaraneni
50 Points
4 years ago
let us consider that cotA,cotB,cotC are in AP and prove that a2,b2,c2 are in AP.
we know that cotA=\frac{b^{2}+c^{2}-a^{2}}{4(area of triangle)} and similarly cotB,cotC..
by considering that cotA,cotB,cotC are inAP,we get
2(c2+a2-b2)=(b2+c2-a2)+(b2+a2-c2),
by simplifying,we get
2b2=a2+c2,
\thereforea2,b2,c2 are in AP  

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