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luni, 21 martie 2022

About the PROBLEM #2 ("икки") from International Olympiad "TUYAMAADA" 2021 - Junior League

           See link, obtained from www.molympiad.net.


          We will formulate the following related problem:


 

"The bisector of angle B of a parallelogram ABCD meets itd diagonal AC at E, and the external bisector of angle B meets line AD at F.

Prove that the right segments CE and FM are parallel and congruent  if and only if BC=2\cdot AB."

 

Proof CiP of Original Problem 

          First of all, notice that from the equal angles:


\angle B'BF\equiv \angle BFA, and \angle CBE \equiv \angle AGC (as angles formed by the parallel lines BC and AD with the transversals BF and BG respectively). And because the bisectors form equal angles, we have \angle B'BF \equiv \angle FAB, \angle CBG \equiv \angle ABG, so we get some isosceles triangles, from where it come

AB=AF=AG\;.

          We will solve the original problem by the vector method.

 Denote with L=BC,\;l=AB. The bisector theorem BE in the triangle ABC

\frac{AE}{EC}=\frac{AB}{BC}=\frac{l}{L} 

allows us to find out  

\overrightarrow{BE}=\frac{L\cdot \overrightarrow{BA}+l\cdot \overrightarrow{BC}}{L+l}.\tag{1}

(Indeed, L\cdot \overrightarrow{AE}=l\cdot \overrightarrow{EC}, or L(\overrightarrow{BE}-\overrightarrow{BA})=l(\overrightarrow{BC}-\overrightarrow{BE})...)

We also obtain from the bisector theorem AE=\frac{l}{L+l}\cdot AC, so \overrightarrow{AE}=\frac{l}{L+l}\overrightarrow {AC}=\frac{l}{L+l}(\overrightarrow{BC}-\overrightarrow{BA}), hence

\overrightarrow {AE}=\frac{l}{L+l}\overrightarrow{BC}-\frac{l}{L+l}\overrightarrow{BA}.

      Further, because AF=AB, we have \overrightarrow{AF}=-\frac{l}{L}\cdot \overrightarrow{AD}=-\frac{l}{L}\overrightarrow{BC}, and now we calculate \overrightarrow{EF}=\overrightarrow{AF}-\overrightarrow{AE}=-\frac{l}{L}\overrightarrow{BC}-(\frac{l}{L+l}\overrightarrow{BC}-\frac{l}{L+l}\overrightarrow{BA}), so 

\overrightarrow{EF}=-\frac{2Ll+l^2}{L(L+l)}\overrightarrow{BC}+\frac{l}{L+l}\overrightarrow{BA}.\tag{2}

 Also \overrightarrow{CM}=\overrightarrow{BM}-\overrightarrow{BC}=\frac{1}{2}\overrightarrow{BE}-\overrightarrow{BC}\;\overset{(1)}{=}\frac{L}{2(L+l}\overrightarrow{BA}+\frac{l}{2(L+l)}\overrightarrow{BC}-\overrightarrow{BC}, getting

\overrightarrow{CM}=-\frac{2L+l}{2(L+l)}\overrightarrow{BC}+\frac{L}{2(L+l)}\overrightarrow{BA}.\tag{3}

           From relations (2) and (3) we see that

\overrightarrow{EF}=\frac{l}{L}[-\frac{2L+l}{L+l}\overrightarrow{BC}=\frac{l}{L+l}\overrightarrow{BA}]=\frac{l}{L}\cdot 2\overrightarrow{CM}, so

\overrightarrow{EF}=\frac{2l}{L}\overrightarrow{CM}.\tag{4}

Relation (4) states that  EF\;\parallel\;CM.

\blacksquare

     REMARK CiP

      Also from the formula (4) we deduce that 

EF \;\overset{\parallel}{=}\;CM\;\;\Leftrightarrow\;\;2l=L\;\;\Leftrightarrow\;\;EFMC-parallelogram\;\;\Leftrightarrow\;\;CE\;\overset{\parallel}{=}\;MF

and we get our reformulation of the problem.

\blacksquare\;\blacksquare

 

 

 

 

 

 

 

 

 


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