In △ ABC, ab = BC, AB is the height of BC side, AE is the bisector of ∠ BAC, ead = 48 ° to find ∠ ACD

In △ ABC, ab = BC, AB is the height of BC side, AE is the bisector of ∠ BAC, ead = 48 ° to find ∠ ACD

Let ∠ ACD = x °,
Because AB = BC
So ∠ BAC = ∠ ACD = x °
Because AE is the bisector of ∠ BAC
So, CAE = 1 / 2, BAC = 1 / 2x degree
Therefore, AED = CAE + ACD = 3 / 2x degree
Because ad is the height of BC side
Therefore, ead + AED = 90 degree
That is 48 ° + 3 / 2x ° = 90 °
x=28

As shown in the figure, ad is the height of △ ABC, make ∠ DCE = ∠ ACD, intersect the extension line of ad at point E, and point F is the symmetrical point of point c about the straight line AE, connecting AF. (1) prove: CE = AF; (2) take a point n on the line AB, make ∠ ENA = 12 ∠ ace, and en intersect BC at point m, connecting am. Please judge the quantitative relationship between ∠ B and ∠ MAF, and explain the reason

(1) It is proved that: ∵ ad is the height of △ ABC, ∵ ADC = ∵ EDC = 90 °, ∵ DCE = ∵ ACD, ∵ ace is isosceles triangle, ∵ AC = CE, and ∵ point F is the symmetry point of point c about AE, ∵ AF = AC, ∵ AF = CE; (2) ∵ B = ∵ MAF, ∴∠1=∠3，∵AC=AF，∴∠4=∠ACD，∵∠ENA=12∠ACE，∠DCE=∠ACD=12∠ACE，∴∠ACD=∠ENA，∴∠4=∠ENA，∵∠4=∠1+∠MAF，∠ENA=∠3+∠B，∴∠B=∠MAF．

As shown in the figure, point D is a point on the edge ab of an equilateral triangle, ab = 3aD, De is perpendicular to point E, AE, CD and intersects point F. prove that ACD of triangle is equal to BAE of triangle

∵AB=3AD,∴AD=2BD；
If △ ABC is an equilateral triangle, then ∠ a = ∠ B = ∠ C = 60 ° and de ⊥ BC, then BD = 2be, then ad = be;
In △ ACD and △ BAE, ∠ Abe = ∠ CAD = 60 °, ad = be, ab = AC, then △ ACD ≌ △ BAE

As shown in the figure, in △ ABC, ab = AC, the center line BD on AC divides the perimeter of the triangle into two parts of 15cm and 12cm, and calculates the length of each side of the triangle

Two possibilities:
1. When two waist AB = AC > bottom BC:
The conditions are: ab + ad = 15
BC+CD=12 ②
∵ BD is the center line of AC side
∴AD=CD
① - 2
AB-BC=3
AC=AB=BC+3 ③
① + 2
AB+BC+(AD+CD)=27
AB+BC+AC=27
2AB+BC=27
Substituting (3) into (3), we get:
2(BC+3)+BC=27
BC=7
That is, the length of the bottom edge is 7
2. When AB = AC