Curt from Reigate College has a neat way of proving this result by rotating one of the triangles about C, keeping the other one fixed, and then using similar triangles.

It is illuminating to see different methods and you may also like to try proving this result using the Sine Rule.

Here is Curt's method.
The question specifies the construction of a line perpendicular to $AB$ drawn through $C$ , and continued until it intersects $DE$ .

It will be proven that the perpendicular to $AB$ through $C$ bisects $DE$ .

To prove this we first develop a useful tool. Note that $∠ BCE = π / 2$ , thus if we were to rotate $B$ by $π / 2$ radians counter-clockwise to $B'$ , then ECB' would be a straight line. Also we note that $∠ ACD = π / 2$ . If we were to rotate $A$ by $π / 2$ counter clockwise to $A'$ , then $A'$ would coincide with $D$ as $| CD | = | CA | = | CA' |$ .

The diagram shows the results of rotating triangle

ABC

π / 2

counter clockwise.

As lengths between points are invariant under such rotations, $| CA | = | CA' |$ , $| AB | = | A' B' |$ and $| BC | = | B' C |$ thus $A' B' C$ is the same triangle as $ABC$ . Clearly $EDC$ is unaffected by the rotation; none of its vertices were rotated or translated.

Now in order to see how this is useful, we start from the first diagram, and draw in the extended perpendicular from $Y$ on $AB$ passing through $C$ intersecting $DE$ at $X$ . We repeat the rotation about $C$ as shown in the next diagram.

In addition to the other transformations already discussed,

Y

goes to

Y'

XCY

is a straight line perpendicular to

AB

, and

Y

is rotated

π / 2

about

C

Y'

. Thus

XCY'

is a right angle and

A' B'

is parallel to

XC

Again, as the distances between points that are rotated in the same manner are invariant under rotation, it follows that $| B' Y' | = | BY |$ and $| A' Y' | = | AY |$ .

It is out intention to show that $| DX | = | EX |$ , or $| DE | = 2 | DX |$

Clearly

ECX

is similar to

EBD'

CX

is parallel to

DB'

thus

∠ EB' D = ∠ ECX

. Similarly angle

∠ EXC = ∠ EDB'

. Both triangles share

∠ DEB'

. Thus all three angles in the two triangles are equal, thus the two triangles are similar.

By the properties of similar triangles, $| EB' | / | EC | = | ED | / | EX |$ . But $| EB' | = 2 | EC |$ . Thus $ED = 2 | EX |$ , as required.