# Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.22.4" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px"]We have a rectangle whose sides are mirrors. A ray of light enters from one of vertices of the rectangle and after being reflected several times, exits via the vertex opposite to the initial one. Prove that at some point of time, the ray passed through the centre of rectangle (Intersection of the diagonals.)

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.23.3" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.23.3"]Do you really need a hint? Try it first![/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.23.3"]

Instead of reflecting the ray and keeping the rectangle fixed, reflect the rectangle and keep the ray fixed.   [/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.23.3"]

Following hint 1, you will get a grid of rectangles and a straight line representing the path of light. If this straight line passes through one of the reflections of the opposite vertex, then in our original representation it has to pass through the opposite vertex. Similarly, if it passes through one of the reflections of the centre then in the original representation, it has to pass through the centre.[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.23.3"] Show that the reflections of $(a,b)$ are of the form $((2m-1)a,(2n-1)b)$. Also, the reflections of the centre $(a/2,b/2)$ are of the form $((p+1/2)a,(q+1/2)b$.[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.23.3"]Suppose that the line indeed passes through a reflection of the opposite vertex. Then it is of the form $x(t)=(2m+1)at, y(t)=(2n+1)bt$. Taking $t=\frac{1}{2}$, we see that it passes through $((m+1/2)a,(n+1/2)b)$, which is a reflection of the centre.[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]

# Similar Problems

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