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 odd if its square is odd

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

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

Proof: 

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

Let us now prove the contrapositive. Since \(n \) is an even number, we can write it in the form \(n = 2k \) where \(k \) is an integer. By squaring this equation we get: \[n ^ 2 = (2k) ^ 2 \] \[n ^ 2 = 4k ^ 2 \] \[n ^ 2 = 2 \left(2k ^ 2 \right) \] Since \(k \) is an  integer then \(2k ^ 2 \) must be an integer due to the closedness of multiplication and exponentiation operations on the set of integers. Denote \(2k ^ 2 \) by \(r \), then we have \(n ^ 2 = 2r \), from which we conclude that \(n ^ 2 \) is an even 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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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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