1, (1) Because AE is perpendicular to BE and CF is perpendicular to AE, triangle BDE and triangle CDF are right-angled triangles with two right angles, a pair of vertex angles are equal, D is the midpoint, and DB=DC, so triangle BDE and triangle CDF are congruent, so BE=CF..(2) If BE = CF is known, because AE. So triangle BDE and triangle CDF are right-angled triangles, two triangles have two right angles respectively, and a pair of vertex angles are equal, so triangle BDE and triangle CDF are congruent, so BD=DC, so D is the midpoint of BC.

2, 123 Push 4 Because AB=AC, AD=AE, AE is perpendicular to BE, AD is perpendicular to DC, and triangle ABE and triangle ACD are congruent, so ∠B=∠C, (assuming that CD and BE intersect at point O), ∠AMD=∠BMO, ∞ So ∠BMO=∠CNO, so ∠AMD=∠ANE, in triangle ADM and triangle AEN, there is a right angle, AD=AE, so these two triangles are congruent, so AM=AN.