As shown in the figure, AC bisector angle bad, CE is perpendicular to ab. (2) when angle ADC + angle ABC = 180 °, 2ae = AB + ad is proved

As shown in the figure, AC bisector angle bad, CE is perpendicular to ab. (2) when angle ADC + angle ABC = 180 °, 2ae = AB + ad is proved

prove:
Make the vertical line of AD through C and cross the extension line of ad to F
It is known that ∠ ADC + ∠ ABC = 180 degree
However, ADC + CDF = 180 degree
Therefore, ABC = CDF
That is, EBC = FDC
Known CE ⊥ ab
Therefore, DFC = BEC = 90 degree
Because AC bisects ∠ bad, CE ⊥ AB, CF ⊥ AF
So CE = CF (the distance from the point on the bisector to both sides of the angle is equal)
So triangle CDF is equal to triangle CBE (AAS)
So EB = FD (the corresponding sides of congruent triangles are equal)
In ACF and ace of right triangle
AC=AC,CE=CF
So right triangle ACF is equal to right triangle ACE (HL)
So AF = AE (the corresponding sides of congruent triangles are equal)
AE = ab-eb, AF = AD + DF
So AE + AF = ab-eb + AD + DF
AF = AE, EB = DF
So 2ae = AB + ad
So AD + AB = 2ae