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# Constructing a Regular Hexagon from Triangles

## Question

In the figure, triangles ABC, DEF, and GHJ are congruent isosceles triangles. Angles B and C are 80 degrees. Points AEFC, EHJD, DFB, and GHF are collinear. Without drawing a circle, show how to mark the corners of a perfect hexagon on the diagram, using only a compass set to a constant length. Prove it's a perfect hexagon.

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I have a solution if I can use a ruler as well as the compass, and the compass is set to the length of BC. Also my solution uses the parallel postulate.

Since E-H-J and G-H-F, angle EHF = angle GHJ = 80o. So the triangle EFH also has base angles of 80o, which makes it isosceles. Similarly, the triangle CBF is also isosceles, with base angles of 80o. The top angles must be 20o accordingly. Now, angle HFB = angle HFE + angle EFB = 20o + (180o - angle EFD) = 20o + 100o = 120o, which is the internal angle of a perfect hexagon.

Centering the compass at B, find the point M on the segment BA so that |BM| = |BC| (which is the set length of the compass). Consider the triangle MBF. Clearly |BM| = |BC| = |BF|, so it is isosceles, with top angle MBF = angle MBC - angle FBC = 80o - 20o = 60o. An isosceles triangle with a top angle of 60o is equilateral. It follows that M is the midpoint of the perfect hexagon with B,F,H as three of the corners. With a ruler, extend the lines BM-, FM-, and HM-. Using the compass centered at M, find the other three corners on those lines.

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Cool. I didn't find the initial 120o angle.

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set compass 1/3 of length HF and strike a point M on FD from point F using compass three times like a divider. This gives equilateral triangle FHM. set compass on F, H, and M and strike the two corresponding segments of equilateral triangle FHM gives the six points of a hexagon. This method still requires a straight edge to connect FH and FM. Not sure from the OP if that is allowed. Or if striking an arc would be allowed (vs drawing a circle). If no ruler is allowed, I cannot visualize where a hexagon might be much less locate it.

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yeah, i took it for granted that we could draw a straight line...

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