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Harmonic mean and constant determination puzzle (Posted on 2015-06-13) Difficulty: 3 of 5
In a triangle ABC , ∠BAC = 90o. Point D lies on the side BC, and satisfies
∠BDA = k*∠BAD, where k is a real constant.

Find the value of k, given that length of AD is the harmonic mean between the respective lengths of BD and CD.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution Solution | Comment 1 of 2
B=(-1,0), C=(1,0), D=(d,0), O=(0,0)
BD=1+d, CD=1-d, OD=d
The harmonic mean is 2(1+d)(1-d)/((1+d)+(1-d))=1-d^2
OA = 1

Using the law of cosines on triangle ADO find 

Using the law of cosines on triangle ABD find 

Using the law of cosines on triangle ABD again

Using the double angle formula for this last cosine 
which is precisely cos(∠BDA)

so k=2

  Posted by Jer on 2015-06-15 12:36:54
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