[

The ratio of the areas of similar triangles ] [ Trigonometric ratios in a right-angled triangle ]

From a point $P$ located inside an acute-angled triangle $A B C$, perpendiculars are dropped to the sides $A B, B C$ and $C A$. The perpendiculars are respectively equal to $l, m, n$. Calculate the area of triangle $A B C$, if the angles $BAC, ABC$ and $ACB$ are respectively $\alpha, \beta$ and $\gamma$.

#
Draw lines through point $P$ parallel to the sides of triangle $ABC$. They divide triangle $ABC$ into six parts, three of which are triangles. If their areas are $S_{1}, S_{2}, S_{3}$, then $S_{\Delta \mathrm{ABC}}=\left(\sqrt{S_{1}}+\sqrt{S_{2}}+\sqrt{S_{3}}\right)^{2}$.

## Solution

Draw lines through point $P$ parallel to the sides of triangle $ABC$. They divide triangle $ABC$ into six parts, three of which are triangles with angles $\alpha, \beta, \gamma$ and heights $l, m, n$. If the areas of these triangles are $S_{1}, S_{2}$, and $S_{3}$ respectively, then

$$
S_{\triangle \mathrm{ABC}}=\left(\sqrt{S_{1}}+\sqrt{S_{2}}+\sqrt{S_{3}}\right)^{2}
$$

Since

$$
\begin{gathered}
S_{1}=\frac{l^{2}(\operatorname{ctg} \alpha+\operatorname{ctg} \beta)}{2}=\frac{l^{2} \sin (\alpha+\beta)}{2 \sin \alpha \sin \beta}=\frac{l^{2} \sin \left(180^{\circ}-\gamma\right)}{2 \sin \alpha \sin \beta}=\frac{l^{2} \sin \gamma}{2 \sin \alpha \sin \beta}=\frac{l^{2} \sin ^{2} \gamma}{2 \sin \alpha \sin \beta \sin \gamma} \\
S_{2}=\frac{m^{2} \sin \alpha}{2 \sin \gamma \sin \beta}=\frac{m^{2} \sin ^{2} \alpha}{2 \sin \gamma \sin \beta \sin \alpha}, S_{3}=\frac{n^{2} \sin \beta}{2 \sin \alpha \sin \gamma}=\frac{n^{2} \sin \beta}{2 \sin \alpha \sin \gamma \sin \beta}
\end{gathered}
$$

then

$$
S_{\Delta \mathrm{ABC}}=\left(\sqrt{S_{1}}+\sqrt{S_{2}}+\sqrt{S_{3}}\right)^{2}=\frac{(l \sin \gamma+m \sin \alpha+n \sin \beta)^{2}}{2 \sin \alpha \sin \beta \sin \gamma}
$$

## Answer

![](https://cdn.mathpix.com/cropped/2024_05_06_e3b1ddf947c7dd2094eeg-20.jpg?height=357&width=832&top_left_y=1994&top_left_x=613)

$\frac{(l \sin \gamma+m \sin \alpha+n \sin \beta)^{2}}{2 \sin \alpha \sin \beta \sin \gamma}$.