The hypotenuse of a right-angled triangle is C. Find the sum of the squares of its medians.

By the property of the median of a right-angled triangle, BB1 = AC / 2.

BB1 ^ 2 = AC ^ 2/4.

From the triangle CC1B, by the Pythagorean theorem, we determine the median CC1.

CC1 ^ 2 = BC ^ 2 + (AB / 2) ^ 2.

From the triangle АА1В, according to the Pythagorean theorem, we determine the median АА1.

AA1 ^ 2 = AB2 + (BC / 2) ^ 2.

Then the sum of the squares of the medians is:

AA1 ^ 2 + BB1 ^ 2 + CC1 ^ 2 = AB ^ 2 + BC ^ 2/4 + AC ^ 2/4 + BC ^ 2 + AB ^ 2/4 = AC ^ 2 + (1/4) * (AB ^ 2 + BC ^ 2) + AC2 / 4 = AC ^ 2 + AC ^ 2/4 + AC ^ 2/4 = 3 * AC ^ 2/2 = 1.5 * AC ^ 2.

Answer: The sum of the squares of the medians is 1.5 * AC ^ 2.