MathDB

The points

D

and

E

lie on the side

(BC)

of the triangle

ABC

such that

D

is between

B

and

E.

A point

R

on the segment

(AE)

is called remarkable if the lines

PQ

and

BC

are parallel, where

\{P\}=DR \cap AC

and

\{Q\}=CR \cap AB.

A point

R'

on the segment

(AD)

is called remarkable if the lines

P'Q'

and

BC

are parallel, where

\{P'\}=BR' \cap AC

and

\{Q'\}=ER' \cap AB.

a) If there exists a remarkable point on the segment

(AE),

prove 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 ratio.

Problem Statement