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Given that x, y, and z are positive real numbers representing the sides of a triangle opposite to angles X, Y, and Z respectively, and the equation tan(X/2) + tan(Z/2) = 2y/(x + y + z) holds true, which of the following statements is correct?

1. ZY equals X + Z
2. Y is the sum of X and Z
3. tan(X/2) equals x divided by (y + z)
4. x² plus z² minus y² equals x times z

Correct answer: Y is the sum of X and Z

Solution

The equation tan(X/2) + tan(Z/2) = 2y/(x + y + z) implies a specific relationship between the angles and sides of the triangle. This relationship confirms that angle Y is the sum of angles X and Z, satisfying the triangle angle sum property.

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