Here's a similar idea I think:

123789^2 + 561945^2 + 642864^2 = 242868^2 + 323787^2 + 761943^2

Take the first, third and fifth digits of each number.

We have: 138^2 + 514^2 + 626^2 = 226^2 + 338^2 + 714^2

Take the second, fourth and sixth digits of each number.

We have: 279^2 + 695^2 + 484^2 = 488^2 + 277^2 + 693^2

Drop the middle two digits of each number.

We have: 1289^2 + 5645^2 + 6464^2 = 2468^2 + 3287^2 + 7643^2

Drop the middle two digits again.

We have: 19^2 + 55^2 + 64^2 = 28^2 + 37^2 + 73^2

We can drop the second and third digits.

We have: 1789^2 + 5945^2 + 6864^2 = 2868^2 + 3787^2 + 7943^2

and drop the second and third digits again.

We have: 19^2 + 55^2 + 64^2 = 28^2 + 37^2 + 73^2

Drop the leftmost digit of each number repeatedly ("beheading"):

23789^2 + 61945^2 + 42864^2 = 42868^2 + 23787^2 + 61943^2

3789^2 + 1945^2 + 2864^2 = 2868^2 + 3787^2 + 1943^2

789^2 + 945^2 + 864^2 = 868^2 + 787^2 + 943^2

89^2 + 45^2 + 64^2 = 68^2 + 87^2 + 43^2

9^2 + 5^2 + 4^2 = 8^2 + 7^2 + 3^2

Drop the rightmost digit of each number repeatedly ("curtailing"):

12378^2 + 56194^2 + 64286^2 = 24286^2 + 32378^2 + 76194^2

1237^2 + 5619^2 + 6428^2 = 2428^2 + 3237^2 + 6719^2

123^2 + 561^2 + 642^2 = 242^2 + 323^2 + 761^2

12^2 + 56^2 + 64^2 = 24^2 + 32^2 + 76^2

1^2 + 5^2 + 6^2 = 2^2 + 3^2 + 7^2

Drop the exponents from all of the preceding equations.

The equalities still hold!