In any triangle ABC $A^{2}$ , $B^{2}$ , $C^{2}$ are in $Q P$ prove that $c o t A$ , $c o t B$ , $c o t C$ ,are in $A . P$

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Solution

Let us consider that cotA,cotB,cotC are in AP and prove that a²,b²,c² are in AP.

we know that cotA= 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,
a²,b²,c² are in AP

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In any triangle ABC A2,B2,C2 are in QP prove that cotA,cotB,cotC,are in A.P

Let us consider that cotA,cotB,cotC are in AP and prove that a2,b2,c2 are in AP.we know that cotA= and similarly cotB,cotC..by considering that cotA,cotB,cotC are inAP,we get2(c2+a2-b2)=(b2+c2-a2)+(b2+a2-c2),by simplifying,we get2b2=a2+c2,a2,b2,c2 are in AP

In any triangle ABC $A^{2}$ , $B^{2}$ , $C^{2}$ are in $Q P$ prove that $c o t A$ , $c o t B$ , $c o t C$ ,are in $A . P$