Arii (Spania OM, 2014)

[mihai miculita]

Fie $ABC$ un triunghi oarecare si punctele arbitrare $D,\,E$ si $F$ ale laturilor $[AB],\,[BC]$ si respectiv $[AC].$ Notam cu $P,\,Q$ si $R$ mijloacele segmentelor $[AE],\,[BF]$ si respectiv $[CD].$ Aratati ca are loc urmatoarea relatie areolara: $\boxed{S_{DFE}=4.S_{PQR}}.$

[mihai miculita]

Folosind coordonate baricentrice, avem: $A(1;0;0),\,B(0;1;0),\,C(0;0;1);$ iar daca:
$D(r;1-r;0),\,E(0;p;1-p),\,F(1-q;0;q),$ atunci:
$R\left(\dfrac{r}{2};\dfrac{1-r}{2};\dfrac{1}{2}\right);$ $P\left(\dfrac{1}{2};\dfrac{p}{2};\dfrac{1-p}{2};\dfrac{1}{2}\right)$ si $Q\left(\dfrac{1-q}{2};\dfrac{1}{2};\dfrac{q}{2};\dfrac{1}{2}\right),$ asa ca:
$\dfrac{S_{D\,EF}}{S_{ABC}}=$$\left|\begin{array}{ccc} r&{1-r}&0\\ 0&p&{1-p}\\ {1-q}&0&q\end{array}\right|=1-p-q-r+pq+pr+rq;\,\,\,(1)$
iar:
$\dfrac{S_{PQR}}{S_{ABC}}=$$\left|\begin{array}{ccc} {\dfrac{1}{2}}&{\dfrac{p}{2}}&{\dfrac{1-p}{2}}\\ {\dfrac{1-q}{2}}&{\dfrac{1}{2}}&{\dfrac{q}{2}}\\ {\dfrac{r}{2}}&{\dfrac{1-r}{2}}&{\dfrac{1}{2}}\end{array}\right|=$$=\dfrac{1}{8}.\left|\begin{array}{ccc} 1&p&{1-p}\\ {1-q}&1&q\\ r&{1-r}&1\end{array}\right|=$
$=\dfrac{1}{4}.(1-p-q-r+pq+pr+rq).\,\,\,(2)$
In fine, din relatiile (1) si (2), rezulta concluzia.