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jlacosta   0
May 1, 2025
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0 replies
jlacosta
May 1, 2025
0 replies
Divisibilty...
Sadigly   10
N 11 minutes ago by Frd_19_Hsnzde
Source: My (fake) translation error
Find all $4$ consecutive even numbers, such that the square of their product divides the sum of their squares.
10 replies
Sadigly
Yesterday at 3:47 PM
Frd_19_Hsnzde
11 minutes ago
Geometry Parallel Proof Problem
CatalanThinker   5
N 16 minutes ago by Tkn
Source: No source found, just yet, please share if you find it though :)
Let M be the midpoint of the side BC of triangle ABC. The bisector of the exterior angle of point A intersects the side BC in D. Let the circumcircle of triangle ADM intersect the lines AB and AC in E and F respectively. If the midpoint of EF is N, prove that MN || AD.
I have done some constructions, but still did not quite get to the answer, see diagram attached below
5 replies
CatalanThinker
Yesterday at 3:33 AM
Tkn
16 minutes ago
help me solve this problem. Thanks
tnhan.129   2
N 21 minutes ago by tnhan.129
Find f:R+ -> R such that:
(x+1/x).f(y) = f(xy) + f(y/x)
2 replies
tnhan.129
6 hours ago
tnhan.129
21 minutes ago
Inequality, inequality, inequality...
Assassino9931   7
N 21 minutes ago by sqing
Source: Al-Khwarizmi Junior International Olympiad 2025 P6
Let $a,b,c$ be real numbers such that \[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]Find the smallest possible value of $a^2 + b^2 + c^2$.

Binh Luan and Nhan Xet, Vietnam
7 replies
1 viewing
Assassino9931
6 hours ago
sqing
21 minutes ago
A robust fact about 5-cycles
math_explorer   2
N May 2, 2016 by math_explorer
[quote]Lemma 3. (Barrington, 1986) There are two five-cycles $\sigma_1$ and $\sigma_2$ in $S_5$ whose commutator is a five-cycle. (The commutator of $a$ and $b$ is $aba^{-1}b^{-1}$.)

Proof. $(12345)(13542)(54321)(24531) = (13254).$[/quote]

Someday I want to write a paper and include a lemma with a proof like this.

Uh, is it just me or is this proof actually incorrect...? I keep getting $(14352)$. I think Mr. Barrington composed his permutations the wrong way. (Fortunately for complexity theory, the lemma is robust to this issue! :P)
2 replies
math_explorer
May 2, 2016
math_explorer
May 2, 2016
No more topics!
Symmedian line
April   91
N Apr 29, 2025 by BS2012
Source: All Russian Olympiad - Problem 9.2, 10.2
Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.
91 replies
April
May 10, 2009
BS2012
Apr 29, 2025
Symmedian line
G H J
Source: All Russian Olympiad - Problem 9.2, 10.2
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April
1270 posts
#1 • 8 Y
Y by Amir Hossein, dave_mathsguru, 799786, itslumi, Adventure10, Mango247, fuzzball2023, and 1 other user
Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.
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mathVNpro
469 posts
#2 • 6 Y
Y by Olemissmath, Psyduck909, Adventure10, Mango247, fuzzball2023, ehuseyinyigit
Let $ M_b$ be midpoint of $ AC$.
We have $ \angle FAM_b = \angle FBC$, $ \angle AM_bF\equiv \angle DM_bF = \angle DEF = \angle BEF = \angle BCF$
$ \Rightarrow \triangle FAM_b\sim \triangle FBC$, $ (i)$. Thus, through the spiral similarity with center $ F$, denote this $ f$ then:
$ f: \triangle FAM_b\mapsto \triangle FBC$
$ \Rightarrow f: A\mapsto B$, $ M_b\mapsto C$
$ \Rightarrow f: A\mapsto M_b$, $ B\mapsto C$
$ \Rightarrow f: \triangle FAB\mapsto \triangle FM_bC$, $ (1)$
But it is easy to notice that: $ \triangle BAF\sim \triangle BM_bC$ (a.a), $ (2)$
$ (1),(2)\Rightarrow \triangle CM_bB\sim \triangle FM_bC$
$ \Rightarrow \frac {CB}{CF} = \frac {CM_b}{FM_b} = \frac {M_bB}{M_bC}$, $ (*)$.
With the same argument, we get $ \triangle BAM_b\sim \triangle BFC$
$ \Rightarrow \triangle BAM_b\sim \triangle AFM_b$ (thanks to $ (i)$).
$ \Rightarrow \frac {BA}{AF} = \frac {M_bB}{M_bA}$, $ (**)$
But $ M_bA = M_bC \Rightarrow \frac {BA}{BC} = \frac {FA}{FC}$ (Combine $ (*)\&(**)$)
$ \Longrightarrow ABCF$ is a harmonic quadrilateral. Therefore, $ BF$ passes through the Pole of $ AC$ wrt $ (O)$, where $ O$ is the circumcenter of $ \triangle ABC$. Then we will get the result of the problem.
Our proof is completed then :lol:
This post has been edited 1 time. Last edited by mathVNpro, May 11, 2009, 4:56 AM
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silouan
3952 posts
#3 • 6 Y
Y by ultralako, Adventure10, Mango247, fuzzball2023, and 2 other users
Let $ M$ the midpoint of $ AC$ , $ O$ the circumcenter of $ \Omega$ and let the tangent to $ \Omega$ at $ B$ meets $ AC$ at $ R$ .
Then $ BOMR$ is cyclic (let $ Q$ its circumcenter) . Then $ R,Q,O$ are collinear . So if the circumcircle of $ BOMR$ meets $ \Omega$ at $ F'$ , then $ BF'\bot RO$ . So by polars $ RF'$ is tangent to $ \Omega$ . By well known theorem we have that $ BF'$ is the $ B$ symmedian. Finally if $ BE$ meets the circumcircle of $ BOMR$ at $ S$ , then $ SM=SF$ . But also because $ RF'=RB$ we have that $ D$ is the incenter of $ F'BM$ so $ SD=SM=SF$ and because $ DM\bot ME$ we also have $ SD=SM=SF=SE$ . This shows that $ MDF'E$ is cyclic and so $ F\equiv F'$ and we are done .
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yetti
2643 posts
#4 • 4 Y
Y by Adventure10, Mango247, and 2 other users
External bisector of $ \angle B$ cuts $ CA$ at $ Y.$ $ I, I_b$ are incenter and B-excenter of $ \triangle ABC.$ $ (E)$ is a circle with center $ E$ and radius $ EA = EC = EI = EI_b.$ The cross ratio $ (B, D, I, I_b)$ is harmonic and $ E$ the midpoint of $ II_b$ $ \Longrightarrow$ $ \overline{BD} \cdot \overline{BE} = \overline{BI}\cdot \overline{BI_b}$ $ \Longrightarrow$ $ BY \perp BDE$ is radical axis of $ \omega, (E).$ Pairwise radical axes $ EF, CA, BY$ of $ \Omega, \omega, (E)$ meet at their radical center $ Y.$ Since $ DF \perp EFY$ and $ BY \perp BDE,$ $ BDFY$ is cyclic. Its circumcircle is the B-Apollonius circle of $ \triangle ABC$ $ \Longrightarrow$ $ BF$ is the B-symmedian.
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Virgil Nicula
7054 posts
#5 • 6 Y
Y by Amir Hossein, Adventure10, Mango247, fuzzball2023, and 2 other users
PP. Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$ . The line $ BD$ intersects the circumcircle $ \Omega$ of
$\triangle ABC$ at $ B$ and $ E$ . Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$ .


Lemma. Let $ ABC$ be a triangle with the circumcircle $ w$ . Denote the following points : the middlepoint $ M$ of the side $ [BC]$ ;

the point $ D\in (BC)$ for which $ \angle DAB\equiv\angle DAC$ ; the point $ E$ for which $ \{A,E\} = AD\cap w$ ; the $ A$-symmedian $ AL$ ,

where $ L\in (BC)$ ; the point $ S$ for which $ \{A,S\} = AL\cap w$ . Then $ SD\perp SE$ , i.e. the quadrilateral $ MDSE$ is cyclically.


Proof. Denote $ XX$ - the tangent in $ X\in w$ to $ w$ . Denote $ T\in BB\cap CC$ . Observe that $ \overline {MET}\perp BC$ and $ \angle ASB\equiv\angle ACB$ .

From the first well-known property, $ T\in \overline {ALS}$ and the line $ \overline{ALST}$ is the $ A$-symmedian in $ \triangle ABC$ $ \Longrightarrow$ $ \left\|\begin{array}{c} \angle BAS\equiv\angle MAC \\
\ \angle SAE\equiv\angle DAM\end{array}\right\|$ .

From the second well-known property, the division $ \{A,S;L,T\}$ is harmonically. Since $ ML\perp MT$ results $ \angle DMA\equiv\angle DMS$ .

Show easily that $ \triangle BAS\sim\triangle MAC$ (a.a.) . Thus, $ \frac {AB}{AS} = \frac {AM}{AC}$ , i.e. $ AS\cdot AM = AB\cdot AC$ .

From the third well-known property $ AD\cdot AE = AB\cdot AC$, obtain $ \boxed {AS\cdot AM = AD\cdot AE = AB\cdot AC}$ , i.e. $ \frac {AS}{AE} = \frac {AD}{AM}$ .

Since $ \angle SAE\equiv\angle DAM$ obtain $ \triangle SAE\sim\triangle DAM$ and in consequence, $ \angle DMA\equiv\angle SEA$ . Since $ \angle DMA\equiv\angle DMS$

and $ \angle SEA\equiv\angle SED$ obtain $ \angle DMS\equiv\angle SED$ , i.e. the quadrilateral $ MDSE$ is cyclically. In conclusion, $ SD\perp SE$ .

Remark. Here are another interesting metrical relations : $ \left\|\begin{array}{ccc} ABS\sim CMS & \implies & \boxed {SB\cdot SC = SA\cdot SM} \\
 \\
AMC\sim CMS & \implies & \boxed {MA\cdot MS = MB^2}\end{array}\right\|$ .

The proposed problem is an immediate consequence of the above lemma.
See PP13 from here
This post has been edited 4 times. Last edited by Virgil Nicula, Aug 11, 2014, 1:54 AM
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jayme
9792 posts
#6 • 3 Y
Y by AlastorMoody, Adventure10, fuzzball2023
Dear Mathlinkers,
1. Let A' be the midpoint of BC, X, Y the second meetpoint of AA', EA' with the circumcircle of ABC and Te the tangent to this circumcircle at E.
2. Te // DA'
3. The circle with diameter ED passes through A'
4. By a converse of Reim's theorem applied to this circle and the circumcircle, FD goes through Y;
5. By a converse of Pascal's theorem applied to EAXFYTe, FX //DA' (BC)
and we are done...
Sincerely
Jean-Louis
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Agr_94_Math
881 posts
#7 • 3 Y
Y by Adventure10, Mango247, ehuseyinyigit
Very nice application of Reim's Theorem jayme.Though a very obvious result, it has great application in many synthetic geometry results.
Thanks.
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mathVNpro
469 posts
#8 • 2 Y
Y by Adventure10, Mango247
jayme wrote:
Dear Mathlinkers,
1. Let A' be the midpoint of BC, X, Y the second meetpoint of AA', EA' with the circumcircle of ABC and Te the tangent to this circumcircle at E.
2. Te // DA'
3. The circle with diameter ED passes through A'
4. By a converse of Reim's theorem applied to this circle and the circumcircle, FD goes through Y;
5. By a converse of Pascal's theorem applied to EAXFYTe, FX //DA' (BC)
and we are done...
Sincerely
Jean-Louis
Do you have any document about Reim's theorem, Jayme? I am so excited about this. :blush:
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jayme
9792 posts
#9 • 3 Y
Y by Adventure10, Mango247, fuzzball2023
See for example my website
http://perso.orange.fr/jl.ayme then: A propos
You will see all the possibility.
Sincerely
Jean-Louis
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ma 29
40 posts
#10 • 16 Y
Y by MexicOMM, Wizard_32, Durjoy1729, HolyMath, Siddharth03, Greenleaf5002, myh2910, snakeaid, centslordm, Jalil_Huseynov, Adventure10, Mango247, GustavoPudim, fuzzball2023, and 2 other users
Dear friends.
I have a short solution for the nice problem,here :) .
//cdn.artofproblemsolving.com/images/4a3755e6df266a6313f50f6a8a6c7c8845715061.jpg

Denote by $ A'$ the midpoint of segment $ AC$.

$ EO$ meets $ (O)$ again at $ P$

By ${ \angle {DFE} = 90^0}$ ,we have $ F,D,P$ are collinear.

From $ BPEF$ is cyclic and $ BDA'P$ is cylic ,we have :

$ \angle {EBF} = \angle {EPF} = \angle{DBA'}$.

Therefor,$ BF$is the symmedian line of the triangle $ BAC$.
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armpist
527 posts
#11 • 3 Y
Y by Adventure10, Mango247, and 1 other user
All are very nice solutions indeed, I must admit.



T.Y.
M.T.
This post has been edited 1 time. Last edited by armpist, May 11, 2009, 8:29 PM
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Virgil Nicula
7054 posts
#12 • 2 Y
Y by Adventure10, Mango247
After the Ma29's shortest proof, this problem seems very simple ...
In the another proofs appeared many nice properties of this configuration.
Thanks to all for their interest and proofs.
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ma 29
40 posts
#13 • 2 Y
Y by Adventure10, Mango247
Hello my friends.
I think all proof above are very nice. :)
Here is another solution to this problem:
//cdn.artofproblemsolving.com/images/c51085c9ea4036fa17fa17def72321254472bf4e.jpg

Denote $ V = EF \cap AC$.

We have :$ \bar {VD} .\bar{VA'} = \bar{VF}.\bar{VE} = \bar{VA}.\bar{VC}$ (1)

But $ A'$ is the midpoint of segment $ AC$.(2)

From (1) ,(2) and Maclaurin's theorem , we obtain the division $ {VDAC}$ is harmonically,so $ V,B,P$ are collinear.

Denote $ Q = BC\cap {PA}$ and denote by $ H$ the intersection of two tangent to $ (O)$ at $ A,C$ ,respectively.

We have $ Q,D,H$ lie on the polar of $ V$ wrt $ (O)$,i.e $ Q,D,H$ are collinear.

By Pascal's theorem applied to six point $ A,A,P,C,B,F$ ,we obtain $ B,F,H$ are collinear.

Therefore ,$ BF$ is the symedian line of the triangle $ ABC$.
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bravado
19 posts
#14 • 2 Y
Y by Adventure10, Mango247
It can also be viewed as an easy application of : http://www.mathlinks.ro/viewtopic.php?p=1082899#1082899

Indeed, from the above problem it follows that the line $ FM$ (where $ M$ is the midpoint of $ AC$) meets the circumcircle of $ \triangle{ABC}$ at a point $ T$ such that $ BT||AC$. By symmetry,
\[ \angle{MBC} = \angle{MTA} = \angle{ABF}.
\]
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Virgil Nicula
7054 posts
#15 • 2 Y
Y by Adventure10, Mango247
We can use my found property at the proposed problem from here.
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