let o be a point inside a triangle ABC such that angle OAC=angle OCB= angle OBA= alpha .then prove that

1)cota +cot b + cot c=cot alpha
2) cosec2a+cosec2b+cosec2c= cosec2alpha

Dear Student,
Please find below the solution to the asked query:

We haveOAB=OBC=OCA=αNowIn AOC, CAO=A-OAB=A-αUsing this in AOC we get COA=π-AIn COA Using sine rule we havesinA-αOC=sinCOAAC sinA-αOC=sinπ-Ab sinA-αOC=sinAb...iSimilarly in COB, suing sine rule you getsinαOC=sinCOBasinαOC=sinπ-CasinαOC=sinCa...iii/iisinA-αsinα=asinAbsinCBy sine rule in ABC, a=ksinA and b=ksinBsinA-αsinα=ksinA.sinAksinB.sinCsinA-αsinα=sinA.sinπ-B+CsinB.sinC=sinA.sinB+CsinB.sinCsinA.cosα-cosA.sinαsinα=sinA.sinB.cosC+sinC.cosBsinB.sinCsinA.sinB.sinC.cosα-sinB.sinC.cosA.sinα=sinA.sinB.cosC.sinα+sinA.sinC.cosB.sinαDivide through out sinA.sinB.sinC.sinαcotα-cotA=cotC+cotBcotA+cotB+cotC=cotαNote: Point Ois known as Brocard point.Now try second part yourself and get back to us in case you face problem.

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