Problem of the Week #71: Monday November 11th, 2024
As before, these problems are the results of me following my curiosity, and I make no promises regarding the topics, difficulty, solvability of these problems.
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This problem is inspired by last week’s problem.
Every line going through the center of a square bisects its area.
a). Does every line going through the center of an equilateral triangle bisect its area too?
b). What is the biggest difference between the areas of the two regions when an equilateral triangle is divided by a line through its center?
c). If we draw every line bisecting the area of an equilateral triangle and mark the point on the line closest to the center, what shape would we trace with these points?
d). For which regular polygons do all lines through its center bisect its area?
e). What is the biggest difference between areas of the two regions when a regular n-gon is divided by a line through its center?
f). If we draw every line bisecting the area of a regular n-gon and mark the point on the line closest to the center, what shape would we trace with these points?
g). Share your own problem inspired by this one.
h). Give one of these questions to a friend/colleague/student/family member to start a mathematical discussion.