The point (m, n) moves on the straight line ax + by + 2C = 0, where a, B, C are the three sides of a right triangle, and C is the hypotenuse, then the minimum value of M2 + N2 is______ ．

The point (m, n) moves on the straight line ax + by + 2C = 0, where a, B, C are the three sides of a right triangle, and C is the hypotenuse, then the minimum value of M2 + N2 is______ ．

According to the meaning of the question, when (m, n) moves to the perpendicular foot position of the origin and the known line, the value of M2 + N2 is the minimum, and the triangle is a right triangle, and C is the hypotenuse. According to the Pythagorean theorem, C2 = A2 + B2, so the distance d from the origin (0, 0) to the line ax + by + 2C = 0 = | 0 + 0 + 2C | A2 + B2 = 2, then the minimum value of M2 + N2 is 4

It is known that a, B and C are the lengths of three sides of a right triangle and C is the hypotenuse. If the point (m, n) is on the straight line ax + by + 2C = 0, then the minimum value of M & sup2; + n & sup2

4
M & sup2; + n & sup2; is the square of the distance from the point on the line ax + by + 2C = 0 to the origin. The minimum value of M & sup2; + n & sup2; is the square of the minimum distance from the origin to the line ax + by + 2C = 0. It is a good condition to use the formula of the distance from the point to the line ("a, B, C are the lengths of the three sides of a right triangle, C is the hypotenuse")

It is proved that a right triangle is a right triangle if its side lengths are a = m square = n square, B = 2 Mn, C = m square + n square (M > n)

"A = m square = n square" should be: "a = m ^ 2-N ^ 2", right?
prove:
because
a^2+b^2
=(m^2-n^2)^2+(2mn)^2
=m^4-2m^2n^2+n^4+4m^2n^2
=m^4+2m^2n^2+n^4
=(m^2+n^2)^2
=c^2
Therefore, according to the inverse theorem of Pythagorean theorem, we know that this triangle is a right triangle
For reference! Jswyc