Explanation:
1) In the figure, as AB is equal to AE, ABE is an equilateral triangle.
As AP is perpendicular to BE
=> AP is the height of the triangle ABE.
In an equilateral triangle, the median and the height is the same, so that AP is also the median of the triangle.
=> P is the midpoint of BE
=> PE = PB
2) In the figure, as AC = AD, so that ACD is an equilateral triangle.
As AP is perpendicular to BE, so that it is perpendicular to CD as well
=> AP is the height of the triangle ACD
In an equilateral triangle, the median and the height is the same, so that AP is also the median of the triangle ACD.
=> P is the midpoint of CD
=> PC = PD
We have:
+) PB = PE
+) PC = PD
=> PB - PC = PE - PD => BC = DE