1, MH: HB = 1: 4 (this can be easily obtained from the similarity in Rt△MCB).

2. Then use Pythagorean theorem to get MH= 1/√5. For convenience, let MH be the unit length a, so MB = 5a.

3.∠ AHM = 45 (this point applies the conclusion of the second question, which shows that ∠AHM=∠BAC, not much explanation) is very useful!

4. After a, do AZ⊥BM at Z (then Z is on the BM extension line). Using information 3, it is known that △AZH is isosceles Rt△.

5. Since AZ // CH and M are AC midpoint and M is ZH midpoint, ZH=2a=ZA, AH=2√2 a, ZB=ZM+MB=6a.

Then there are three situations:

1, the simplest one: AD=AH=2√2 a=2√ 10/5.

2、AD=DH:

Connecting ZD, it is found that △ZAD and △ZHD are congruent (S.S.S congruence), so ZD is the bisector of ∠AZB angle.

So ad: db = az: zb = 2a: 6a = 1: 3.

And AB=√2 AB = 2√2, so AD= AB/ 4 = √2/2.

3、AH=HD:

MX⊥AB in X exceeds M, and HY⊥AB in Y exceeds H.

From MX // HY: hy: MX = HB: MB = 4a: 5a = 4: 5.

In isosceles Rt△AXM, AM= 1, so MX = √2/2, so HY = 2 √2/5.

And AH = 2√2 a = 2√2/√5, and it is found that Rt△AYH has the structure of 1: 2: √ 5 (it can be written directly by Pythagorean theorem), so AY=2 HY = 4 √2/5.

In isosceles △AHD, HY is the AD midline of the base, so AY=YD, so AD= 2 AY = 8 √2/5.

In this case, your denominator is wrong.

Add a few points:

1, I think this is relatively simple.

2. Is there an angular bisector theorem in the junior high school syllabus? I don't remember. If not, just check. The basic theorem of plane geometry is still very important and easy to use.

As a college student, I may not be as detailed as I was in Grade Three, but I hope you can understand.

I'll come up with some useful information first. You can also use this format when writing a certificate, or you can put useful information into the corresponding three situations step by step to prove it. Logically, the latter is clearer, but the overall grasp of the former is more hierarchical, depending on which one you adapt to. To be on the safe side, you'd better use the former, because I don't remember the grading standards of that year. You may lose points if you use the latter, so you have to take the exam.

As a candidate in Zhabei, I wish you a smooth college entrance examination.