MathDB

Let

ABC

be a fixed acute-angled triangle. Consider some points

E

and

F

lying on the sides

AC

and

AB

, respectively, and let

M

be the midpoint of

EF

. Let the perpendicular bisector of

EF

intersect the line

BC

K

, and let the perpendicular bisector of

MK

intersect the lines

AC

and

AB

S

and

T

, respectively. We call the pair

(E, F )

<span class='latex-italic'>interesting</span>

, if the quadrilateral

KSAT

is cyclic. Suppose that the pairs

(E_1 , F_1 )

and

(E_2 , F_2 )

are interesting. Prove that

\displaystyle\frac{E_1 E_2}{AB}=\frac{F_1 F_2}{AC}

Proposed by Ali Zamani, Iran

G6