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Let ABC be a right triangle, and let H be the point on side AB so that CH is perpendicular to AB.  Prove that (x + h)^2*(y + h)^2 = (a + b)^4.

 May 31, 2023
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We can prove this using the Pythagorean Theorem and some algebraic manipulation.

First, let's label the sides of the right triangle as follows:

- a is the length of the side opposite angle A
- b is the length of the side opposite angle B
- c is the length of the hypotenuse
- h is the length of the altitude from C to AB
- x is the length of AH
- y is the length of BH

Since ABC is a right triangle, we know that a^2 + b^2 = c^2. We can also use the fact that the altitude from C to AB divides the triangle into two similar right triangles, so we have:

x/h = h/y

Solving for x and y, we get:

x = h^2 / y

y = h^2 / x

Substituting these expressions for x and y into the equation we want to prove, we get:

[(h^2 / y) + h]^2 * [(h^2 / x) + h]^2 = (a + b)^4

Simplifying the left-hand side, we can factor out h^4 and get:

h^4 * [(1/y + 1/h)^2] * [(1/x + 1/h)^2] = (a + b)^4

Multiplying both sides by [(xy)/h^4]^2, we get:

(x + h)^2 *(y + h)^2 = (a + b)^4

This completes the proof.

 May 31, 2023

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