## BDMO NATIONAL Junior 2016/04

Posts: 29
Joined: Mon Jan 23, 2017 10:32 am

### BDMO NATIONAL Junior 2016/04

please give the solution in picture .i can't understand the writting
Attachments Screenshot_2017-01-29-19-18-16-2.png (15.24 KiB) Viewed 2782 times

Posts: 181
Joined: Mon Mar 28, 2016 6:21 pm

### Re: BDMO NATIONAL Junior 2016/04

This is known as the perpendicular lemma. It is quite handy in proving perpendicularity.
PROOF

Let $AC\cap BD=E$. Now, apply the Law of Cosines on triangles $ABE$,$CBE$,$CDE$,$DAE$. Let $\angle AEB=\theta$. We have
\begin{aligned} BA^2&=AE^2+BE^2-2(AE)(BE)(\cos \theta)\\ BC^2&=CE^2+BE^2+2(BE)(CE)(\cos \theta)\\ DC^2&=DE^2+CE^2-2(DE)(CE)(\cos \theta)\\ AD^2&=AE^2+DE^2+2(AE)(DE)(\cos \theta)\\ \end{aligned}

and substituting into $AB^2-AD^2=BC^2-CD^2$ yields that $\cos\theta=0$. So, $\theta=90$.
Frankly, my dear, I don't give a damn.

Posts: 29
Joined: Mon Jan 23, 2017 10:32 am

### Re: BDMO NATIONAL Junior 2016/04

Posts: 181
Joined: Mon Mar 28, 2016 6:21 pm

### Re: BDMO NATIONAL Junior 2016/04

I'm afraid the other solution I know is with vectors.
Frankly, my dear, I don't give a damn.

dshasan
Posts: 66
Joined: Fri Aug 14, 2015 6:32 pm

### Re: BDMO NATIONAL Junior 2016/04

Let's assume $AC$ is not perpendicular to $BD$.

WLOG, lets assume $\angle AOB < 90$. Let $X$ be the foot of the perpendicular from $B$ to AC and let $Y$ be the foot of the perpendicular from $D$ to $AC$.

Now, extension of Pythagporous theorem gives

$BO^2 + AO^2 + 2.AO.OX + DO^2 + CO^2+ 2.CO.OY = BO^2 + OC^2 - 2.OC.OX + DO^2 + AO^2 - 2AO.OY$

Or, $2.AO.OX + 2.CO.OY = -(2.OC.OX+2.AO.OY)$

Or, $2AO(OX+OY) = -2.OC(OX+OY)$

But that is only possible when $OX = OY = 0$, or $X,Y$ lies on $O$. So, the diagonals $AC,BC$ are perpendicular to each other.
The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.

- Charles Caleb Colton

protaya das
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Joined: Fri Nov 24, 2017 9:41 pm

### Re: BDMO NATIONAL Junior 2016/04 plz say what is perpendicullar lemma. i am protaya das. a winner of national bdmo on 2017 from junior category

Posts: 181
Joined: Mon Mar 28, 2016 6:21 pm

### Re: BDMO NATIONAL Junior 2016/04

protaya das wrote:
Sun Dec 10, 2017 3:34 pm plz say what is perpendicullar lemma. i am protaya das. a winner of national bdmo on 2017 from junior category
Hello Protya Das, winner of bdmo 2017, the perpendicular lemma is this
Attachments Screenshot from 2018-02-17 01-49-58.png (299.7 KiB) Viewed 2450 times
Frankly, my dear, I don't give a damn.

prottoydas
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Joined: Thu Feb 01, 2018 11:56 am

### Re: BDMO NATIONAL Junior 2016/04

hello protaya and prottoy das are the same person.I have forgot the password of protaya das so i am using the prottoy das id.

prottoy das
Posts: 17
Joined: Thu Feb 01, 2018 11:28 am
Location: Sylhet

### Re: BDMO NATIONAL Junior 2016/04

the problem is from the book euclidean geometry in mathmatical olympiad