MHB Prove Angle of Diagonals in Quadrilateral is Degrees

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The discussion focuses on proving that the diagonals of a quadrilateral formed by the intersection points of a square's sides with a strip intersect at a specific angle. The problem presents an algebraic solution but seeks a geometric approach for deeper understanding. Key insights include the relationship between angles α and β, which sum to 90 degrees, and the properties of the isosceles triangles KNP and MLQ formed in the configuration. The angles at the intersection points can be expressed in terms of α and β, leading to the conclusion that the angle γ is 45 degrees. This geometric perspective enhances comprehension of the angle relationships in the quadrilateral.
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Two angles of a square with side
c7d457e388298246adb06c587bccd419ea67f7e8.png
protrude beyond a strip of width
c7d457e388298246adb06c587bccd419ea67f7e8.png
with parallel edges. The sides of the square intersect the edges of the strip at four points. Prove that the diagonals of the quadrilateral whose vertices are these points intersect at an angle of
f27da4fca6f3fecb215974023aad210b70ec3857.png
degrees.
rys.png
 

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As in a previous problem, here is an algebraic solution. I would like to see a geometric solution giving more insight into why this result holds.
We may as well take $a=1$. Tilt the diagram so that the square becomes the unit square, and suppose that the parallel lines make an angle $\theta$ with the $x$-axis.

[TIKZ][scale=1.25]
\draw [very thick] (0,0) -- (4,0) -- (4,4) -- (0,4) -- cycle ;
\draw [very thick] (-4,0.5) -- (4,6.5) ;
\draw [very thick] (0,-1.5) -- (8,4.5) ;
\coordinate [label=left:$A$] (A) at (0,3.5) ;
\coordinate [label=above:$B$] (B) at (0.667,4) ;
\coordinate [label=right:$C$] (C) at (4,1.5) ;
\coordinate [label=below:$D$] (D) at (2,0) ;
\draw [very thick] (A) -- (C) ;
\draw [very thick] (B) -- (D) ;
\draw (2.5,0.2) node {$\theta$} ;
\draw (1.3,2.6) node {$\phi$} ;
\draw (-0.5,0) node {$(0,0)$} ;
\draw (4.5,0) node {$(1,0)$} ;
\draw (-0.5,4) node {$(0,1)$} ;
\draw (4.5,4) node {$(1,1)$} ; [/TIKZ]
The equations of the parallel lines are then of the form $y = x\tan\theta + c+d$ and $y = x\tan\theta + c-d$ for some constants $c$ and $d$. The condition that the lines form a strip of width $1$ is $2d\cos\theta = 1$. So $2d = \sec\theta$.

The points where the lines cross the sides of the square are then $$ A = (0,c+d),\qquad B = ((1-c-d)\cot\theta,1), \qquad C = (1,\tan\theta + c-d), \qquad D = ((d-c)\cot\theta,0) .$$ The slope of the line $AC$ is therefore $\tan\theta - 2d = \tan\theta - \sec\theta$.

The slope of $BD$ is $\dfrac{-1}{(2d-1)\cot\theta} = \dfrac{-\tan\theta}{\sec\theta-1}$.

It follows from the formula $\tan(\alpha - \beta) = \frac{\tan\alpha -\tan\beta}{1+ \tan\alpha\tan\beta}$ that if $\phi$ is the angle between $AC$ and $BD$ then $$\begin{aligned} \tan\phi = \frac{(\tan\theta - \sec\theta) - \frac{-\tan\theta}{\sec\theta-1}}{1 + (\tan\theta - \sec\theta) \frac{-\tan\theta}{\sec\theta-1}} &= \frac{(\sec\theta - 1)(\tan\theta - \sec\theta) + \tan\theta}{(\sec\theta - 1) - \tan\theta (\tan\theta - \sec\theta)} \\ &= \frac{\sec\theta(\tan\theta - \sec\theta + 1)}{\sec\theta(1 - \sec\theta + \tan\theta)} \\ &= 1. \end{aligned}$$ It follows that $\phi = 45^\circ$.
 
Hint
156973.png

156973.png
 
1 / α + β = 90
2 / it is enough to note that the triangles
KNP and MLQ (they have two heights of length "a") so they are isosceles
so |∡NMQ| =90 - α/2 and |∡MNP| =90 - β/2
γ=(α + β)/2=45
 
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