Inscribing hexagon in circle

...

It's easy enough to show using Trigonometry, but that doesn't constitute a geometric proof, which as I recall has to be done with only compass and straight edge. But it's simple enough to demonstrate with a compass and straight edge. I don't know whether demonstration by construction constitutes a formal geometric proof, or not.

Set your compass to a convenient radius and draw a circle. Without changing the compass setting strike an arc from any point on the circle that intersects the circle. From that intersection, strike another intersecting arc. Continue around the circle and, if done carefully enough, the 6th arc will pass through the original point. Since all arcs have the same radius, all the chords connecting the intersections are the same length and equal to the radius of the arc which is also the radius of the circle. Connect each point of intersection with its neighbors using a straight line. By definition,

Tom Veatch Wichita, KS USA

An armed society is a polite society. Manners are good when one may have to back up his acts with his life. Robert A. Heinlein

Given your demonstration, I believe you can safely append:

QED

Regards,

Tom Watson

Remember the first three (3) plane geometry proofs?

Side-Side-Side works for me.

Lew

An equilateral triangle has equal sides and three 60-degree angles. Arrange six of them with sides s in an array with one common vertex, and you will have the inscribed hexagon.

On 5/19/2009 6:47 PM Tom Veatch spake thus:

That's a demonstration, not a proof. But you knew that.

I'm interested in the proof. It can't be all that complicated.

On 5/19/2009 7:02 PM Lew Hodgett spake thus:

Expand, please. Don't know exactly how this proof works.

It is well-known that one cannot trisect an angle with a straight-edge and compass, so I don't think you'll get a proof with that approach. On the other hand, using division you can divide 360 degrees, or 2*Pi radians by 6 to get the angle for each slice of the pie The rest has already been discussed (side-side-side).

Bill

Plug " Side-Side-Side" into Google, should keep you out of trouble for a couple of hours, especially the congruent triangle proofs.

Lew

Or, put another way:

Consider a regular hexagon. Draw line segments from the center of the hexagon to each of the six vertices. These six equal angles at the center must add up to 360, so each is 60. Since the triangles are isosceles, and their angles add to 180, they are also equilateral. So the side of the hexagon is equal to the length of the line form the center to a vertex on the hexagon, which is the radius of the circle.

On 5/19/2009 7:18 PM alexy spake thus:

I don't see any proof in there, only an assertion.

On 5/19/2009 7:42 PM alexy spake thus:

That sounds better. (Don't know if it constitutes a rigorous proof or not, but it satisfies my "itching".)

Look deeper. This is a proof of the "from which it can be clearly seen..." type that occasionally drove me batty!

================================= This is another purely graphical soultion.

The illustration with the A, B, C, D and E circles gives the OP his 3 pie pieces, if he just puts the triangle inside the hexagon.

OK so far...

.. but you just went astray there.

That's not sufficient to prove that the triangles are equilateral, since the angles add to 180 in *all* triangles.

This may be what you meant to say:

Since the angle at the vertex of each triangle is 60 degrees, the sum of the angles at the base is 180 - 60 = 120 degrees. Since each triangle is isosceles, the angles at the base are equal, and (since they add to 120) therefore also 60 degrees. The triangles are therefore equiangular, and therefore equilateral.

Nobody's attempting to trisect an angle in that approach; in fact, it's essentially the same method as Euclid's proof, a link to which was already posted up-thread.

But I also pointed out that the triangle was isosceles. Isosceles and one 60 degree angle are necessary and sufficient conditions for an equilateral triangle (in Euclidean geometry).

That is what I said. You just have to read between the lines .

Well, I havn't seen the proof but dividing a circle into thirds is very easy with a compass. Set your radius, and keep it there. Draw the circle, set the compass point on the circle and draw an arc inside the circle. Repeat 5 times using an intersect as another pivot point. You will get a perfect 6-petal flower.

1. Take the circle, and draw a radius.
2. Use a compass to measure from the intersection of the radius and circle, to the center.
3. Scribe a circle from that point.
4. connect the center to the 2 new intersections
5. You now have 2 equilateral triangles inside the circle (all 3 sides are equal. - they're radii) you now have a third of a circle (or 2 sixths)
6. continue all the way around for the hexagon

shelly