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**mukesh****Member**- Registered: 2010-07-18
- Posts: 30

if the median 'AD' of a triangle 'ABC' is perpendicular to side AB then prove that 'tanA+2tanB=0'

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,311

hi mukesh,

Here's an outline of a way to prove this. see diagram below.

There's no right angle to get tanA easily so I used the sine and cosine rules:

and

Put these together to get tanA and simplify.

work on this expression for tanA, making use of the following:

After much simplification you can get this equal to -2tanB, from which the required result follows.

It's a tough one so expect it to take 2 or 3 pages. If you get stuck post back where you've got to, and I'll compare your answer with mine.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,397

Well, I have a solution that takes 2 or 3 lines to write up, but requires quite a bit of inspection.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,311

hi Stefy,

What does 'quite a bit of inspection' mean exactly.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,397

I do not have an exact definition. Roughly, it means that some times you will think of it, sometimes you won't, and it mostly depends on luck, not unlike many other geometry problems.

My solution extends the line AB and names E the foot of the perpendicular from C to that line. Then I used The basic trig equations to get the result.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,311

hi Stefy,

That is a brilliant way to do it. Short and no complicated trig stuff. I am in awe. ( no dazzled-smiley-face available)

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,397

The thing I dislike about those kinds of ideas are that they can be tough to get. It does come down to luck and experience a lot.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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