**36**
rahul
**·**2012-05-20 14:48:39
can't get ur soln....

either the problem lies with the literature of my question or maybe u r wrong...

here's the exact question with the figure once again......

Given a triangle ABC. A line from A meeting BC at O is drawn.

perpendiculars are drawn on AO from B and C at M and N respectively.

If D is the mid point of BC then prove that,

DM = DN

**11**
sougata nag
**·**2012-06-03 01:30:11

let BC & AN meet at O.

now, Î”MOBâ‰ˆÎ”KODâ‰ˆÎ”OCN

MK/BD=KO/OD=ON/OC=KN/CD. as CD=BD so MK=KN so MD=DN.

[proved]

**36**
rahul
**·**2012-05-21 16:23:43
ya... now that is appreciable...!!

**21**
Shubhodip
**·**2012-05-21 15:19:24
Denote the intersection of MN and BC by O

since BM || NC

ON/ OC = OM/OB = (ON + OM)/(OC + OB) = MN/BC = MF/BD

so OM/OB = MF/BD = (OM- MF)/(OB - BD) = OF/OD

so OF/OD = MF/BD so the result follows by thales

(No construction, nothing )

**36**
rahul
**·**2012-05-21 09:39:23
well my soln. using Apollonious..!!

The red lines are the contructions i've done..!!

Now, using Appolonius theorem...

In triangle ABC we have,

BM^{2} + CM^{2} = 2 (BD^{2} + DM^{2}) --- (i)

And, in triangle BNC we have,

BN^{2} + CN^{2} = 2 (BD^{2} + DN^{2}) ---- (ii)

on (i) - (ii) we have,

BM^{2} - BN^{2} + CM^{2} - CN^{2} = 2 (DM^{2} - DN^{2})

=> - MN^{2} + MN^{2} = 2 (DM^{2} - DN^{2})

=> DM = DN

**36**
rahul
**·**2012-05-21 09:26:13
what's wrong with u??????

In triangle BOM , "D" is not the mid point of OB...!!

How then can u say DF || BM ???????

actually i've already done this question with Apollonius

And here goes the Thales' theorem...

and what u r trying to say is...

Here, F is the mid point of MN , D is the mid point of BC

then which part of Thales' theorem says DF || BM

rather, if D would have been the mid point of A line through N which would meet

BM at some point then one could say, DF || BM...

nd i don't knw what r u trying to do?????

if u can prove it using thales' theorem by constructing triangles and all then do it....

i just want to see the proof....

that's all....!!

**21**
Shubhodip
**·**2012-05-21 06:00:57
(Breaking the promise)

F is the mid point of MN

then BM || DF || NC

What the hell is wrong here ?

Can't you just try to prove BM || DF || NC ?

**36**
rahul
**·**2012-05-21 05:52:54
i think the concept that u r using here is cent % wrong....

nd has no relation with thales' theorem.....

jst look at the statement uv posted..

nd thats wht iv been trying to tell u...

**21**
Shubhodip
**·**2012-05-21 01:36:27
(for the last time )

Let F be the mid point of MN

DF ||BM || NC (if u cant get this by thales, use complex number/coordinate )

so MN perpendicular to DF, F is midpoint so FD is it's perpendicular bisector ; so DM = DN

**21**
Shubhodip
**·**2012-05-17 04:22:25
This is extremely easy, it probably reduces to the fact that , in isosceles triangle ABC, C lies on the perpendicular bisector of AB

**21**
Shubhodip
**·**2012-05-20 06:09:57
F is midpoint of MN, D is mid point of BC

BM || NC

so BM ||NC || DF (use thales)

**158**
Anik Chatterjee
**·**2012-05-20 04:41:31
@shubhodip couldnt get...can u please explain?

@rahul can u give the solution.. y r images not being uploaded properly?? this is annoying ...

**36**
rahul
**·**2012-05-20 00:08:13
well i to have a soln..

did it using Apollonius theorem..!!

**36**
rahul
**·**2012-05-19 23:49:08
what?

how can DF||BM||CN by ur statement?

afterall u can't make the median perpendicular to MN from D unless u prove the statement

either

DM = DN (That's what u need to prove)

or,

that DF bisects angle MDN..

cun't get... explain please..!!

**21**
Shubhodip
**·**2012-05-19 15:26:19
idk if u can see the image, at least I don't

Let F be the midpoint of MN

so DF || BM || CN

so MN perpendicular to DF, F is midpoint so FD is it's perpendicular bisector ; so DM = DN

**36**
rahul
**·**2012-05-19 15:06:34
@Shubhodip - i don't think so... can u show ur work?

@Anik - try to figure out what the statement says... its easy..!!

**158**
Anik Chatterjee
**·**2012-05-18 03:37:07
how can CN be perpendicular to AN??