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Subject: "Morley's Trisector proofs use more ..."     Previous Topic | Next Topic
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mhbrill
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2 posts
Aug-10-01, 11:25 AM (EST)
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"Morley's Trisector proofs use more than angles-why?"
 
   After reading the various proofs of the Morley Trisector Theorem on this site, there still seems to be a miracle. The Morley Trisector Theorem is this: The three points of intersection of the adjacent trisectors of the angles of any triangle form an equilateral triangle.

The Morley construction is done entirely by using angles (input two of the angles of the scalene triangle, trisect all three of these angles, intersect adjacent trisectors, etc.) It'stands to reason that there should be enough relationships among these angles to provide a proof of the theorem. But there are not enough pure angle relationships to solve for all the angles.

There are 15 putatively different angles in the problem, from which the two input scalene angles can be subtracted leaving 13 unknowns. There are eight triangles (each of which has a sum-of-three-angles equation), and three internal vertices (each of which has a sum-of-four-angles equation). That makes 11 equations in the 13 unknowns. One of these equations drops out as an identity, leaving 3 unsolvable unknowns.

All the proofs you cite resort to ratios of distances as well as angles (except Figure 1 in the "behold" proof of D.J. Newman, which ASSUMES, but does not derive, enough angles to solve). Why do angles alone fail to prove the Morley Trisector Theorem?


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alexb
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Aug-14-01, 12:43 PM (EST)
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1. "RE: Morley's Trisector proofs use more than angles-why?"
In response to message #0
 
   >The Morley construction is done entirely
>by using angles (input two
>of the angles of the
>scalene triangle, trisect all three
>of these angles, intersect adjacent
>trisectors, etc.) It'stands
>to reason that there should
>be enough relationships among these
>angles to provide a proof
>of the theorem. But
>there are not enough pure
>angle relationships to solve for
>all the angles.

In proof #1 at

https://www.cut-the-knot.com/triangle/Morley/

one can replace the last identity for the sides of the equilateral triangle with identities for its angles through the theorem of sines, as one posibility.

In this manner the sides play an auxiliary role, nothing else.

Your question is somewhat ambiguous. Do you consider the law of sines a mixed (side/angle) identity, or an angle identity? If the latter, your question has been answered above. If the former, then you should rethink your question, because specifying the angles fixes the shapes of the triangle and leads to the law of sines.


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