Proof Concerning Central Line \(X_5X_6\) of Triangle

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Hello everyone. Today we will prove theorem about central line \(X_5X_6\) of triangle. Proof of the Central Line in a Golden Rectangle Construction Statement of the Theorem Let \(ABCD\) be a golden rectangle where \( \frac{AB}{BC} = \phi \), and construct the square \( BCQP \) inside it. Reflect \( P \) over \( D \) to obtain \( E \). Then, the line \( EB \) coincides with the central line \( X_5X_6 \) of triangle \( ABP \). Step-by-Step Proof 1. Define the Coordinates We assign coordinates as follows: \( A = (0,0) \), \( B = (\phi x, 0) \), \( C = (\phi x, x) \), \( D = (0, x) \). The square \( BCPQ \) ensures \( P = (\phi x, 2x) \). The reflection of \( P \) across \( D \) is \( E = (-\phi x, 2x) \). 2. Compute the Nine-Point Center \( X_5 \) The nine-point center \( X_5 \) is the circumcenter of the medial triangle, which consists of the midpoints: \[ M_1 = \left(\frac{0 + \phi x}{2}, 0\right) = \left(\frac{\phi x}{2}, 0\right), \] \[ M_2 = \left(\f...
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Proof that an integer is even if its square is even

 Hello everyone. Today we will prove that an integer is even if its square is even. We will use proof by contraposition as proof method.

Theorem: Let \(n \) be an integer. If \(n ^ 2 \) is an even number, then \(n \) is also an even number. 

Proof: 

The contrapositive of this statement is: Let \(n \) be an integer. If \(n \) is an odd number, then \(n ^ 2 \) is also an odd number. 

Let us now prove the contrapositive. Since \(n \) is an odd number, we can write it in the form \(n = 2k + 1 \) where \(k \) is an integer. By squaring this equation we get: \[n ^ 2 = (2k + 1) ^ 2 \] \[n ^ 2 = 4k ^ 2 + 4k + 1 \] \[n ^ 2 = 2 \left(2k ^ 2 + 2k \right) +1 \] Since \(k \) is an integer then \(2k ^ 2 + 2k \) must be an integer due to the closedness of addition, multiplication and addition operations on the set of integers. Denote \(2k ^ 2 + 2k \) by \(l \), then we have that \(n ^ 2 = 2l + 1 \), from which we conclude that \(n ^ 2 \) is an odd number. Since we have proved that the contrapositive of the original statement is correct, it  means that the original statement must be correct as well. 

\(\blacksquare \)

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