Estonian Mathematical Olympiad

Denote the area of the figure $K$ by $S_K$. Note (Fig. 25) that $$S_{A{B}^{\prime }C{A}^{\prime }B{C}^{\prime }}=S_{AP{B}^{\prime }}+S_{B^{\prime }PC}+S_{CP{A}^{\prime }}+S_{A^{\prime }PB}+S_{BP{C}^{\prime }}+S_{C^{\prime }PA}.$$ Furthermore, $S_{AP{B}^{\prime }}=S_{APB}$, because triangles $AP{B}^{\prime }$ and $APB$ have the same altitude drawn from vertex $A$ and the corresponding bases are equal. Similarly we get $S_{B^{\prime }PC}=S_{BPC}$, $S_{CP{A}^{\prime }}=S_{CPA}$, $S_{A^{\prime }PB}=S_{APB}$, $S_{BP{C}^{\prime }}=S_{BPC}$ and $S_{C^{\prime }PA}=S_{CPA}$. Hence $$S_{A{B}^{\prime }C{A}^{\prime }B{C}^{\prime }}=2S_{APB}+2S_{BPC}+2S_{CPA}=2(S_{APB}+S_{BPC}+S_{CPA})=2S_{ABC}.$$

Fig. 25