98. Symmedian & median (metrical relations).
by Virgil Nicula, Sep 3, 2010, 4:54 AM
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=364921
Proof. Observe that
, 
, i.e. 
, 
and 
. Using similarities 
, 
and 
. Observe that 
.
Observe that
![$\left[2(b^2+c^2)-a^2\right]\left[(b^2+c^2)+4b^2c^2\right]=a^2(b^2+c^2)^2-4a^2b^2c^2$](//latex.artofproblemsolving.com/6/2/a/62aa3b924eecd41e695d3d0419a90b7fac056524.png)
.
Demonstratie. Notam 
-mediana ![$[AM]$](//latex.artofproblemsolving.com/1/f/9/1f9b22599237fb6240a50b5f75e8f6ced1292374.png)
si -simediana ![$[AS]$](//latex.artofproblemsolving.com/8/1/f/81f842d87b22553e4b674bb2bf46a080da785009.png)
, unde 
. Presupunem fara a restrange generalitatea ca 
. Se arata usor ca 
si 
. Asadar 
este bisectoarea unghiului 
daca si numai daca 
.
Observatie. Notam proiectia
a lui pe 
. Se arata usor ca 
.
Quote:
In 
, the -median meets at 
and the circumcircle again at 
. The -symmedian meets at 
and the circumcircle again at 
. Prove that 
and 
.
, the 
-median meets 
at 
and the circumcircle again at 
. The -symmedian meets at 
and the circumcircle again at 
. Prove that 
and 
.Proof. Observe that
, 
, i.e. 
, 
and 
. Using similarities 
, 
and 
. Observe that 
.Observe that
![$\left[2(b^2+c^2)-a^2\right]\left[(b^2+c^2)+4b^2c^2\right]=a^2(b^2+c^2)^2-4a^2b^2c^2$](http://latex.artofproblemsolving.com/6/2/a/62aa3b924eecd41e695d3d0419a90b7fac056524.png)
.Quote:
Proposed problem. In with the -median and -symmedian meet at and at . Prove that 
.
with 
the -median and -symmedian meet at and at . Prove that 
.Quote:
Generalization. Let 
be a triangle. Consider two points ![$\{M,S\}\subset [BC]$](//latex.artofproblemsolving.com/d/5/b/d5b7a05901ea048585eaddab2294d3933909032e.png)
of so that 
.
The lines
, 
meet again the circumcircle of in , respectively. Then 
and
.
be a triangle. Consider two points ![$\{M,S\}\subset [BC]$](http://latex.artofproblemsolving.com/d/5/b/d5b7a05901ea048585eaddab2294d3933909032e.png)
of so that 
.The lines
, 
meet again the circumcircle of in , respectively. Then 
and
.Quote:
Proposed problem. In denote 
for wich 
. Consider two points
so that
, 
and . Prove that 
.
denote 
for wich 
. Consider two points 
so that
, 
and . Prove that 
.Quote:
Proposed problem. Fie triunghiul in care notam lungimea 
a medianei din varful . Sa se arate ca mediana
si simediana din varful trisecteaza unghiul 
daca si numai daca 
.
in care notam lungimea 
a medianei din varful . Sa se arate ca medianasi simediana din varful
trisecteaza unghiul 
daca si numai daca 
. -mediana ![$[AM]$](http://latex.artofproblemsolving.com/1/f/9/1f9b22599237fb6240a50b5f75e8f6ced1292374.png)
si -simediana ![$[AS]$](http://latex.artofproblemsolving.com/8/1/f/81f842d87b22553e4b674bb2bf46a080da785009.png)
, unde . Presupunem fara a restrange generalitatea ca . Se arata usor ca 
si 
. Asadar 
este bisectoarea unghiului 
daca si numai daca 
.Observatie. Notam proiectia
a lui pe . Se arata usor ca 
.This post has been edited 33 times. Last edited by Virgil Nicula, Nov 23, 2015, 8:34 AM
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