Circle is symetric about diameter

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Zzzz
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Circle is symetric about diameter

Unread post by Zzzz » Thu Feb 17, 2011 8:39 am

Easy to prove but interesting :
$A$ and $B$ are two points on a circle and $C$ is a point on diameter $MN$ such that $\angle ACN=\angle BCN$
Prove that $AC=BC$
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Re: Circle is symetric about diameter

Unread post by Moon » Thu Feb 17, 2011 9:06 am

A reflection is enough! :D
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Re: Circle is symetric about diameter

Unread post by photon » Sun Feb 20, 2011 6:38 pm

add A,B.suppose AB intersects MN at D.
\[\angle ACD=\angle BCD\]
\[\angle ADC=\angle BDC\]
as MN is a line from center to AB,so it is peripendicular .
DC is common side.ADC,BDC ARE SORBOSOMO.AC=BC
@moon bhaiya,what is a reflection (in geometry)?
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Re: Circle is symetric about diameter

Unread post by Moon » Sun Feb 20, 2011 10:53 pm

How did you prove $\angle ADC=\angle BDC$? I don't really understand why "MN is a line from center to AB,so it is peripendicular" implies this.

BTW almost reflection is the same as physics. Here if we assume $MN$ as a mirror then then the reflection of the half circle of the left hand side is the same as the right half of the circle (imagine it in your mind!). Also as $\angle ACN=\angle BCN$, the line $AC$ will be on the line $CB$. Also you see that the points $A,B$ are on the circles, the reflection of $A$ must coincide with $B$. (Actually it is enough to say that we want to reflect $A$ with respect to $MN$, but I have explained everything for you) :)
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Re: Circle is symetric about diameter

Unread post by photon » Mon Feb 21, 2011 5:45 pm

:o ...D is not midpoint :( ,,,but i applied 9-10text book upapadya 35 for that :shock:
here i am giving another solution; confused to say it a solution,it may be wrong also.but with ALL IZZ WELL theory i should exchange my wrong and (sometimes) right thoughts to learn... :geek:
suppose (for the 2nd time), the reflection of A and B are ACP and BCQ.here,
\[\angle ACP=180-2\angle ACN,
\angle BCQ=180-2\angle BCN,\]
\[so,\angle ACP=\angle BCQ\]
in a circle ,if many angles in a point are same,then the CHAPs (which made the angles) will be equal and the chords will be equal.so AP=BQ.
i think in 2 triangles,1angle equals to another of the triangle and their opposite sides are equal,then the will be SORBOSOM.according to it ,CAP and CBQ are 'sorbosom'.AC=BC
please reply about that.is it ok about equal angles and opposite equal sides?i thought about \
on that.
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Re: Circle is symetric about diameter

Unread post by Zzzz » Mon Feb 21, 2011 6:49 pm

photon wrote:the reflection of A and B are ACP and BCQ.
Actually the term 'reflection' is almost same as physics as Moon vai said. So the reflection of a man will be a man, a tree will be a tree, a line will be a line, a point will be a point. Probably you didn't mean reflection here. Please attach a photo (you can find the option in full editor) to clarify the points $P$ and $Q$.
photon wrote:in a circle ,if many angles in a point are same,then the CHAPs (which made the angles) will be equal and the chords will be equal
Mmm... how can we say that? It is not a proved theorem :?
photon wrote:i think in 2 triangles,1angle equals to another of the triangle and their opposite sides are equal,then the will be SORBOSOM.
Its not true actually.

Please don't use any theorem if you are not sure about it. Even if you are sure about it, but you can't give any reference then prove it.
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Re: Circle is symetric about diameter

Unread post by photon » Mon Feb 21, 2011 8:32 pm

actually i meant AC's reflection CP and BC's reflection CQ.
and the triangles (equal angles and equal opposite sides) thought came to my mind on that way.i am really confused about that. :? can anyone show some logic against or behind it?
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Re: Circle is symetric about diameter

Unread post by photon » Tue Feb 22, 2011 9:49 am

edit:i got it.it is not possible.
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Re: Circle is symetric about diameter

Unread post by Abid » Wed Feb 23, 2011 2:57 pm

(It can be wrong, um not sure)
on the pic,

A theke MN aar upor AP lombo abong B theke BQ lombo tani. <CAP= <CBN [90 degree]

<ACP= <BCQ [given]

<CAP= <BCQ [ 90 degree - <ACP = 90 degree- < BCQ]

shutorang, ACP and BCQ shodrish.

akhon, ata prove korle hobe j, P o Q akoi bindu. Tahole CP= CQ hobe. Tahole ACP o BCQ shorboshomo hobe.

Shutorabg AC= BC hobe.

Now can anyone prove that P and Q are same point??
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Re: Circle is symetric about diameter

Unread post by photon » Thu Feb 24, 2011 4:32 pm

i am trying now SONCHARPATH.
if you take any 2 points from a circle, their SONCHARPATH will be a diameter.it is easy to prove.here it is.
we take 2 points X,Y from a circle and find their SONCHARPATH M'N'(a chord of the circle ).add X,Y and take a point F from M'N'.add X,F and Y,F.
XY intersects M'N' at Q.now triangles XQF and YQF are SORBOSOMO(3 sides' equality).we find M'N' is perpendicular bi-sector of XY chord.it can be possible only for diameter.so it is proved that SONCHARPATH OF 2 points of a circle will be a diameter.
it is enough to prove that A,B's SONCHARPATH is MN.......
suppose, A,B's SONCHARPATH is not MN.suppose MN is SONCHARPATH of A,P.P is any point of circumference.
add C,P and A,P.
AP crosses MN at D.
of course,ACD,CDP are SORBOSOMO.(3 sides' equality)
\[\angle ACD=\angle PCD

\]
\[but, \angle ACD=\angle BCD\]
\[SO,\angle BCD=\angle PCD\]
but angle PCD is either greater then angle BCD, or lesser then angle BCD.
so P,B same point.MN is SONCARPATH of A,B.
so AC=BC.
i am pretty sure of that provement. :P
Last edited by photon on Fri Feb 25, 2011 11:22 am, edited 1 time in total.
Try not to become a man of success but rather to become a man of value.-Albert Einstein

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