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 Harmonic mean and constant determination puzzle (Posted on 2015-06-13)
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)

Comments: ( Back to comment list | You must be logged in to post comments.)
 A different approach Comment 2 of 2 |
The harmonic property leads to:   |BD||CD| = |AD||BC|/2

Using the intersecting chord theorem, it follows that the circle
with diameter BC and centre O, passes through A (because
of the right angle) and a point E where ADE is a straight line
with |DE| = |BC|/2, which is the radius of the circle.

/BDA    = /EDO (vert opp)
= /EOD (angles of isosceles triangle EDO)
= 2 /BAD (angles at centre and circumf)
So k = 2

 Posted by Harry on 2015-06-16 10:16:51
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