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The Sum of Any Two Sides of a Triangle Is Greater Than the Third Side

If x, y and z...

Question

If x, y and z are the 3 sides of a triangle, then select the correct ones.

x+y > z

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x-y > z

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y + z > x

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y - z > x

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x+z > y

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z< x+y

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x < y+ z

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y < x+ z

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\begin{matrix} (x + y)^{2} & z x & z y z x & (z + y)^{2} & x y z y & x y & (z + x)^{2} \end{matrix}

2 x y z (x + y + z)^{3}

x (2 x + 3 y - z) + y^{2} (3 x + z - 2 y) - z (2 x + 3 z - y) .

∣ ∣ ∣ ∣ ∣ \begin{matrix} (x + y)^{2} & z x & z y z x & (z + y)^{2} & x y z y & x y & (z + x)^{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ = 2 x y z (x + y + z)^{3}

If x, y and z are variables, verify the cyclic symmetry of the following expressions.

(1) x(y + z) + y(z + x) + z(x + y)

(2) xy(x − y) + yz(y − z) + zx(z − x)

(3) x²y(x + y) + y²z(y + z) + z²x(z + x)

(4) x³(x + y) + y³(y + z) + z³(z + x)

(5) xy²(x − y) + yz²(y − z) + zx²(z − x)

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