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