Hey there! Let's break down this geometry problem step by step to figure out why the perimeter of △ADH is 42.

First, let's restate the given details clearly:

• In △ABC, AB = 18, AC = 24, BC = 30
• BF bisects ∠ABC, CF bisects ∠ACB (so F is the incenter of △ABC)
• DH ∥ BC, with D lying on AB, H lying on AC, and DH passing through point F

The core of solving this problem lies in combining angle bisector properties and parallel line angle relationships to identify isosceles triangles, which gives us equal side lengths we can substitute.

Let's start with the left side of the diagram:

1. Since BF is the angle bisector of ∠ABC, we know ∠ABF = ∠CBF.
2. Because DH ∥ BC, alternate interior angles are equal: ∠DFB = ∠CBF.
3. Putting these two together, ∠ABF = ∠DFB. This makes △DBF an isosceles triangle, so DB = DF.

Now let's look at the right side:

1. CF is the angle bisector of ∠ACB, so ∠ACF = ∠BCF.
2. Again, DH ∥ BC, so alternate interior angles give us ∠HFC = ∠BCF.
3. This means ∠ACF = ∠HFC, making △HCF an isosceles triangle, so HC = HF.

Now let's calculate the perimeter of △ADH. By definition, the perimeter is:
Perimeter = AD + DH + AH

We can split DH into DF + FH, so substitute that into the formula:
Perimeter = AD + DF + FH + AH

Now replace DF with DB (from our first isosceles triangle) and FH with HC (from the second):
Perimeter = AD + DB + HC + AH

Group the terms to match the sides of △ABC:
Perimeter = (AD + DB) + (AH + HC)

But AD + DB is just the length of AB, and AH + HC is the length of AC! So:
Perimeter = AB + AC = 18 + 24 = 42

That's the trick—we don't even need to use the length of BC here, which is a neat shortcut to get the answer quickly.

备注：内容来源于stack exchange，提问作者LL636