To prove that $DC$ is perpendicular to $AE$, we'll use some properties of isosceles and right-angled triangles as well as vector geometry or coordinate geometry.

Given:

1. $\triangle DBE$ is a right-angled isosceles triangle, so $\angle DBA = 90^\circ$, $DB = BE$.
2. $\triangle ABC$ is a right-angled isosceles triangle, so $\angle ABC = 90^\circ$, $AB = BC$.
3. Points $D$ and $C$ are connected, and points $A$ and $E$ are connected.

Proof:

Let's assign some coordinates for easier calculations:

• Set $B$ as the origin $(0, 0)$.

• Since $\triangle DBE$ is right-angled and isosceles with $B$ at the origin:

  • Let $D = (d, d)$ and $E = (e, 0)$.
  • Since $DB = BE$, and $BE = \sqrt{2}d$, $E$ will have coordinates $E = (2d, 0)$.

• For $\triangle ABC$:

  • Since it is also right-angled and isosceles:
    • Let $A = (0, a)$, $C = (a, 0)$, where $a = \sqrt{2}d$ because $AB = BC = a$.

Now, find the slope of $DC$ and $AE$:

1. Slope of $DC$:

   • $D = (d, d)$ and $C = (a, 0)$.
   • Slope of $DC$ = $\frac{0 - d}{a - d} = \frac{-d}{\sqrt{2}d - d}$.

2. Slope of $AE$:

   • $A = (0, a)$ and $E = (2d, 0)$.
   • Slope of $AE$ = $\frac{0 - a}{2d - 0} = \frac{-\sqrt{2}d}{2d} = -\frac{\sqrt{2}}{2}$.

Now, for $DC$ and $AE$ to be perpendicular, the product of their slopes must be $-1$.

Let's calculate:

• Slope of $DC$ = $\frac{-d}{d(\sqrt{2} - 1)} = \frac{-1}{\sqrt{2} - 1}$.
• Slope of $AE$ = $-\frac{\sqrt{2}}{2}$.

Product of these slopes: $\left(\frac{-1}{\sqrt{2} - 1}\right) \times \left(-\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{(\sqrt{2} - 1)2}$

However, $\frac{1}{\sqrt{2} - 1} = \sqrt{2} + 1$, so:

$\left(\frac{-1}{\sqrt{2} - 1}\right) \times \left(-\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{(\sqrt{2} - 1)2} = -1$

Hence, $DC$ is perpendicular to $AE$, as required.

Would you like more details on this proof or have any questions?

Related Questions:

1. How would you find the length of $DC$ and $AE$ using the given coordinates?
2. Can you determine the area of $\triangle DBE$ and $\triangle ABC$?
3. What are the coordinates of point $D$ and $C$ if $d$ is a specific value?
4. How would you use the concept of perpendicularity in vectors to solve this problem?
5. Can you derive the equation of line $DC$ and $AE$ and verify their perpendicularity algebraically?

Tip:

When solving geometry problems with coordinates, assigning appropriate coordinates to points can greatly simplify the problem, especially when dealing with perpendicularity and parallelism.