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Explanation of Elements VI.3

Here is the original statement:

*If an angle of a triangle be bisected and the straight line
cutting the angle cut the base also, the segments of the base will have
the same ratio as the remaining sides of the triangle; and, if the segments
of the base have the same ratio as the remaining sides of the triangle,
the straight line joined from the vertex to the point of section will bisect
the angle of the triangle.*

It always helps to have a diagram:

This looks different from the original diagram used by Euclid, but is
easier to follow when we apply it to Archimedes' method. Euclid's first
claim is that if *AD* bisects angle *BAC*, then

*BD : CD = BA : AC.*
Note that here the "base" is whichever side is opposite the angle being
bisected, so it is not necessarily the bottom; and the "remaining sides"
are the two that delineate the angle being bisected.

For the converse, Euclid is saying that if we cut side *BC* into
two parts, where *D* is the cutting point (the "point of section"),
and we manage to do it so that *BD : CD = BA : AC*, then this
will guarantee that *AD *bisects angle *BAC*. The phrase "point
of section" is common in reading Euclid, and you will encounter it frequently.
It simply refers to the dividing point whenever we divide a segment into
two parts.

One last thing to note is that, although we have drawn the figure here
so that it looks like a right triangle, that's not part of the theorem.
This works for *any* triangle.

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