So the angle ADB=90 degrees

So the triangle ADB is a right triangle.

Because BE is perpendicular to AC and e.

So AEB angle =90 degrees.

So the triangle AEB is a right triangle.

Because m is the midpoint of AB

So em and DM are the center lines of right triangle AEB and right triangle AEB respectively.

So EM= 1/2AB.

DM= 1/2AB

So EM=DM

So the triangle MDE is an isosceles triangle

Because n is the midpoint of DE

So MN is the center line of the isosceles triangle MDE.

So MN is the perpendicular of isosceles triangle MDE (isosceles triangle with three lines in one)

So MN is perpendicular to DE.

(2) Solution: Let AD and BE intersect at O point.

Because angle DOB= angle OAB+ angle OBA

Because angle AEB= angle ADB=90 degrees

So a, b, d and e are four-point circles.

So angle DEB= angle OAB

Because angle C=60 degrees, angle ADC= angle ADB=90 degrees.

Because angle C+ angle ADC+ angle DAC= 180 degrees.

So the angle DAC=30 degrees.

Because angle DAC+ angle AEB+ angle AOE= 180 degrees.

So AOE angle =60 degrees

Because angle AOE= angle DOB

Because m is the midpoint of AB in the right triangle AEB.

So I =MB

So angle MEB= angle MBE

So angle DOB= angle DEB+J MEB=60 degrees.

Because MN is perpendicular to DE (proved)

So the angle MNE=90 degrees.

So tan60=MN/NE

So MN/NE= root number 3

Because n is the midpoint of DE

So NE= 1/2DE

So MN:DE= root number 3:2