Proof Concerning Central Line X5X6X_5X_6 of Triangle

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Hello everyone. Today we will prove theorem about central line X5X6X_5X_6 of triangle. Proof of the Central Line in a Golden Rectangle Construction Statement of the Theorem Let ABCDABCD be a golden rectangle where ABBC=ϕ \frac{AB}{BC} = \phi , and construct the square BCQP BCQP inside it. Reflect P P over D D to obtain E E . Then, the line EB EB coincides with the central line X5X6 X_5X_6 of triangle ABP ABP . Step-by-Step Proof 1. Define the Coordinates We assign coordinates as follows: A=(0,0) A = (0,0) , B=(ϕx,0) B = (\phi x, 0) , C=(ϕx,x) C = (\phi x, x) , D=(0,x) D = (0, x) . The square BCPQ BCPQ ensures P=(ϕx,2x) P = (\phi x, 2x) . The reflection of P P across D D is E=(−ϕx,2x) E = (-\phi x, 2x) . 2. Compute the Nine-Point Center X5 X_5 The nine-point center X5 X_5 is the circumcenter of the medial triangle, which consists of the midpoints: M1=(0+ϕx2,0)=(ϕx2,0), 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 nn be an integer. If n2n ^ 2 is an odd number, then nn is also an odd number. 

Proof: 

The contrapositive of this statement is: Let nn be an integer. If nn is an even number, then n2n ^ 2 is also an even number. 

Let us now prove the contrapositive. Since nn is an even number, we can write it in the form n=2kn = 2k where kk is an integer. By squaring this equation we get: n2=(2k)2n ^ 2 = (2k) ^ 2 n2=4k2n ^ 2 = 4k ^ 2 n2=2(2k2)n ^ 2 = 2 \left(2k ^ 2 \right) Since kk is an  integer then 2k22k ^ 2 must be an integer due to the closedness of multiplication and exponentiation operations on the set of integers. Denote 2k22k ^ 2 by rr , then we have n2=2rn ^ 2 = 2r , from which we conclude that n2n ^ 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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