## Zadatak A-4.2.

Neka je $A B C$ pravokutni trokut s pravim kutom u vrhu $C$. Neka su $A^{\prime}, B^{\prime}, C^{\prime}$ redom nožišta okomica povučenih iz težišta trokuta $A B C$ na pravce $B C, C A, A B$.

Odredi omjer površina trokuta $A^{\prime} B^{\prime} C^{\prime}$ i $A B C$.


## Prvo rješenje.

Označimo sa $T$ težište trokuta $A B C$ te sa $a, b, c$ i $v$ redom duljine stranica $\overline{B C}, \overline{C A}, \overline{A B}$ i visine na stranicu $\overline{A B}$. Označimo i $\alpha=\varangle C A B$ i $\beta=\varangle A B C$.

Budući da je $T$ težište trokuta $A B C$, vrijedi

$$
\left|T A^{\prime}\right|=\left|B^{\prime} C\right|=\frac{1}{3}|A C|=\frac{1}{3} b, \quad\left|T B^{\prime}\right|=\left|A^{\prime} C\right|=\frac{1}{3}|B C|=\frac{1}{3} a, \quad\left|T C^{\prime}\right|=\frac{1}{3} v
$$

![](https://cdn.mathpix.com/cropped/2024_05_30_6f525b6526cae5162cdfg-17.jpg?height=540&width=805&top_left_y=912&top_left_x=634)

Vrijedi

$$
\begin{aligned}
P\left(A^{\prime} B^{\prime} C^{\prime}\right) & =P\left(A^{\prime} B^{\prime} T\right)+P\left(B^{\prime} C^{\prime} T\right)+P\left(C^{\prime} A^{\prime} T\right) \\
& =\frac{1}{2}\left(\left|T A^{\prime}\right| \cdot\left|T B^{\prime}\right|+\left|T B^{\prime}\right| \cdot\left|T C^{\prime}\right| \cdot \sin \left(180^{\circ}-\alpha\right)+\left|T C^{\prime}\right| \cdot\left|T A^{\prime}\right| \cdot \sin \left(180^{\circ}-\beta\right)\right) \\
& =\frac{1}{2} \cdot \frac{1}{9}(a b+a v \sin \alpha+b v \sin \beta)
\end{aligned}
$$

Budući da vrijedi $v=a \sin \beta, v=b \sin \alpha, a=c \sin \alpha, b=c \sin \beta$ i $c^{2}=a^{2}+b^{2}$, imamo:

$$
\begin{aligned}
P\left(A^{\prime} B^{\prime} C^{\prime}\right) & =\frac{1}{18}\left(a b+a^{2} \sin \alpha \sin \beta+b^{2} \sin \alpha \sin \beta\right) \\
& =\frac{1}{18}\left(a b+c^{2} \sin \alpha \sin \beta\right)=\frac{1}{18}(a b+a b) \\
& =\frac{1}{18} \cdot 2 a b=\frac{2}{9} P(A B C)
\end{aligned}
$$

Dakle, dobivamo $P\left(A^{\prime} B^{\prime} C^{\prime}\right): P(A B C)=2: 9$.

