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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