求助：给定直角与边长条件，求证△AED为等腰三角形

Alright, let's work through this problem step by step. You tried using the Pythagorean theorem without much progress—let's switch gears to angle chasing and isosceles triangle properties, which will make this proof straightforward.

First, let's recap our given information clearly:

• ∠ABC and ∠BDE are both right angles (90°)
• The length of BD equals the length of BC (( |BD| = |BC| ))

Step 1: Use Isosceles Triangle Angle Properties #

Since ( |BD| = |BC| ), △BCD is an isosceles triangle. By the rule that angles opposite equal sides are equal:

∠BCD = ∠BDC

Step 2: Leverage Right Triangle Angle Sum #

In the right triangle △ABC (right-angled at B), the two non-right angles add up to 90°:

∠BCD + ∠BAC = 90°

Step 3: Relate Angles Using the Right Angle ∠BDE #

We know ∠BDE = 90°, meaning BD is perpendicular to DE. Looking at the position of point D on AC (with the order A-D-C), ∠BDC and ∠ADE are complementary (they add up to 90°):

∠BDC + ∠ADE = 90°

Step 4: Equate Angles Through Substitution #

From Step 1, we know ∠BCD = ∠BDC. Substitute this into the equation from Step 2:

∠BDC + ∠BAC = 90°

Now compare this to the equation from Step 3: both have ∠BDC plus another angle equal to 90°. If we subtract ∠BDC from both equations, we get:

∠BAC = ∠ADE

Step 5: Conclude △AED is Isosceles #

In △AED, ∠BAC is the same as ∠EAD. Since we just proved ∠EAD = ∠ADE, by the converse of the isosceles triangle theorem (equal angles imply equal opposite sides), we can say:

( |AE| = |DE| )

This means △AED has two equal sides, so it's an isosceles triangle.

If you still want to verify with the Pythagorean theorem, assign variables to the sides (e.g., let ( |BC| = |BD| = x ), ( |AB| = y )) and use coordinate geometry or similar triangles to calculate the lengths of AE and DE—you'll find they're equal.

内容的提问来源于stack exchange，提问作者fehlerteufel123