Question 115
Points $D$ and $E$ are considered on the side ($BC$) of triangle $ABC$, with $D$ between $B$ and $E$.
About a point $R$ of the segment ($AE$) we will say that it is *remarkable* if the lines $PQ$ and $BC$ are parallel, where $\{P\} = DR \cap AC$ and $\{Q\} = CR \cap AB$.
About a point $R'$ of the segment ($AD$) we will say that it is *remarkable* if the lines $P'Q'$ and $BC$ are parallel, where $\{P'\} = BR' \cap AC$ and $\{Q'\} = ER' \cap AB$.
a) If there is a remarkable point on the segment ($AE$), show that any point of the segment ($AE$) is remarkable.
b) If each of the segments ($AD$) and ($AE$) contains a remarkable point, prove that $BD = CE = \varphi \cdot DE$, where $\varphi = \frac{1+\sqrt{5}}{2}$ is the golden number.
About a point $R$ of the segment ($AE$) we will say that it is *remarkable* if the lines $PQ$ and $BC$ are parallel, where $\{P\} = DR \cap AC$ and $\{Q\} = CR \cap AB$.
About a point $R'$ of the segment ($AD$) we will say that it is *remarkable* if the lines $P'Q'$ and $BC$ are parallel, where $\{P'\} = BR' \cap AC$ and $\{Q'\} = ER' \cap AB$.
a) If there is a remarkable point on the segment ($AE$), show that any point of the segment ($AE$) is remarkable.
b) If each of the segments ($AD$) and ($AE$) contains a remarkable point, prove that $BD = CE = \varphi \cdot DE$, where $\varphi = \frac{1+\sqrt{5}}{2}$ is the golden number.