interior angle of heptagon inscribed (2013 UQ/QAMT PS Competition 8.1)

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interior angle of heptagon inscribed (2013 UQ/QAMT PS Competition 8.1)

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#1 h Feb 27, 2021, 9:29 AM

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Consider the star made by connecting $7$ points equally spaced around the circumference of a circle, as shown in the diagram. What is the interior angle of the heptagon?

https://cdn.artofproblemsolving.com/attachments/a/b/f27870e12b775132b2f2f180208302b9fcc3d8.png

This post has been edited 2 times. Last edited by parmenides51, Feb 27, 2021, 9:30 AM

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#2 h Feb 27, 2021, 9:29 AM

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posted for the image link

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#3 h Feb 27, 2021, 2:38 PM

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Is the heptagon not regular? The angles match up.

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#4 h Feb 27, 2021, 3:01 PM

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It's not convex therefore not regular, it's 7 out of 13 vertices are the vertices of a regular heptagon
in fact it is a not convex 13-gon , the word heptagon seems to confuse

on seconds thought it might be considered as a hexagon if you imgagine / let the sides intersect each other,
probably that is what it means

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#5 h Feb 27, 2021, 4:07 PM

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@#3: I believe so.
@#4: Let the be $A, B, C, D, E, F, G.$ Obviously $ABCDEFG$ is regular, and if we are considering the inner heptagon whose vertices are the intersections of the segments, then let the intersection of $\overline{AF}$ and $\overline{EG}$ be $X.$ We have $\angle AXE=\frac{\overarc{ACE}+\overarc{FG}}{2}.$ Now, since our seven points are equally spaced around the circumference of the circle, we have $\angle AXE = \frac{360^\circ \left(\frac{4}{7}+\frac{1}{7}\right)}{2} = \frac{360^\circ \cdot \frac{5}{7}}{2} = \boxed{\frac{900^\circ}{7}}.$ EDIT: If we wish to take the interior acute angle of the star, we can simply compute $180^\circ-2\left(180^\circ-\frac{900^\circ}{7}\right)=180^\circ-2 \cdot \frac{360^\circ}{7} = \boxed{\frac{540^\circ}{7}}.$

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#6 h Feb 27, 2021, 6:44 PM

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Note that $\frac{900}{7}$ is the interior angle of an equilangular polygon. Is the polygon equilangular but not regular?

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#7 h Feb 27, 2021, 7:26 PM

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OlympusHero wrote:

Note that $\frac{900}{7}$ is the interior angle of an equilangular polygon. Is the polygon equilangular but not regular?

I'm suspecting the star is equilateral and the two heptagons (the inner heptagon and the outer heptagon whose sides have not been drawn in the diagram) are regular...

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