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Triangles in a Polygon

Date: 06/14/97 at 19:36:10
From: Krishna
Subject: regular polygons

Dear Dr. Math,

A regular 18-sided polygon is inscribed in a circle and triangles are 
formed by joining any three of the eighteen vertices.  How many obtuse 
triangles are there?

I didn't get too far because an 18-gon is very large.  I tried to get 
some idea by drawing an octagon, but I couldn't get too far with it 

Thank you

Date: 06/16/97 at 19:28:37
From: Doctor Wilkinson
Subject: Re: regular polygons

You started out with a very good idea: if the problem is too big, try
something smaller.  You ran into difficulties because an octagon is 
still pretty big. When you have a good idea, push it! If you try a 
pentagon, you may really be able to see what is going on.

For a pentagon, it's really easy. There are only 5 diagonals, and you 
can spot the obtuse triangles right away: there are just 5 of them.


Now what's happening here? To get an obtuse triangle, the side 
opposite the obtuse angle has to join two vertices with just one 
vertex in between.  And the arc of the circle joining these two 
vertices and passing through the vertex at the obtuse angle must be 
less than half the circle.  If we got a semicircle, we would have a 
right angle, and anything more gives us an acute angle.  So for the 
pentagon, there's only this one possible shape for the triangle, and 
we can rotate it into 5 different positions.

Now we're probably not ready to leap all the way to 18.  Let's try 6.


Now from vertex 1 to vertex 4 is half the circle, which would give us 
a right angle, so again we can only go from vertex 1 to vertex 3, 
which leaves just one vertex in between.  But now there are 6 vertices 
altogether, so we get 6 possible obtuse triangles.

Now let's try your octagon.  


This time, if we go from vertex 1 to vertex 5, that's a right angle, 
so we can't go that far.  But we can go from vertex 1 to vertex 4, and 
now there are two vertices in between, so we get 2 triangles that way.  
And again we have a triangle with the side opposite the obtuse angle 
going from 1 to 3.  So that's 3 shapes of triangle, and we can rotate 
them around to 8 positions, for 24 obtuse triangles altogether.

Do you think you're ready to do 18 now?

-Doctor Wilkinson,  The Math Forum
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Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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