Hi all,
I am studying an analysis of the following lemma:
"Given a circle and two points D, E within it, to draw straight lines thought D, E to a point A on the circumference in such a way that, if they meet the circle again in B, C, BC shall be parallel to DE."
Heath (A History of Greek Mathematics), offers the following analysis.
1. Suppose the problem solved and DA, EA drawn ('inflected') to A in such a way that, if AD, AE meet the circle again in B, C, BC is parallel to DE.
2. Draw the tangent at B meeting ED produced in F.
3. Then angle FBD = angle ACB = angle AED
4. Therefore, A, E, B, F are concyclic
5. and consequently FD.DE = AD.DB
6. But the rectangle AD.DB is given, since it depends only on the position of D in relation to the circle, and the circle is given.
7. Therefore the rectangle FD.DE is given.
8. And DE is given; therefore FD is given, and therefore F.
9. It follows that the tangent FB is given in position, and therefore B is given.
10. Therefore BDA is given and consequently AE also.
11. To solve the problem, therefore, we merely take F on ED produced such that FD.DE = the given rectangle made my the segments of any chord through D, draw the tangent FB, join BD and produce it to A, and lastly draw AE through to C; BC is then parallel to DE.
The following diagram is provided as well.
Now I have two questions:
1) Which further theorems are necessary to deduce step 4 from step 3? The angle at C does not seem to play a role and C is not on the produced circle.
2) I don't understand the notation FD.DE and AD.DB. In particular I am confused that Heath, in step 6 and 7, calls AD.DB and FD.DE "rectangles", since the points A, D and B lie on one straight line and so do the points F,D and E. Since the points all lie on one line, no rectangle is given by them.
I appreciate any hints that could shed light on this question.
All best,
Bphil
I am studying an analysis of the following lemma:
"Given a circle and two points D, E within it, to draw straight lines thought D, E to a point A on the circumference in such a way that, if they meet the circle again in B, C, BC shall be parallel to DE."
Heath (A History of Greek Mathematics), offers the following analysis.
1. Suppose the problem solved and DA, EA drawn ('inflected') to A in such a way that, if AD, AE meet the circle again in B, C, BC is parallel to DE.
2. Draw the tangent at B meeting ED produced in F.
3. Then angle FBD = angle ACB = angle AED
4. Therefore, A, E, B, F are concyclic
5. and consequently FD.DE = AD.DB
6. But the rectangle AD.DB is given, since it depends only on the position of D in relation to the circle, and the circle is given.
7. Therefore the rectangle FD.DE is given.
8. And DE is given; therefore FD is given, and therefore F.
9. It follows that the tangent FB is given in position, and therefore B is given.
10. Therefore BDA is given and consequently AE also.
11. To solve the problem, therefore, we merely take F on ED produced such that FD.DE = the given rectangle made my the segments of any chord through D, draw the tangent FB, join BD and produce it to A, and lastly draw AE through to C; BC is then parallel to DE.
The following diagram is provided as well.
Now I have two questions:
1) Which further theorems are necessary to deduce step 4 from step 3? The angle at C does not seem to play a role and C is not on the produced circle.
2) I don't understand the notation FD.DE and AD.DB. In particular I am confused that Heath, in step 6 and 7, calls AD.DB and FD.DE "rectangles", since the points A, D and B lie on one straight line and so do the points F,D and E. Since the points all lie on one line, no rectangle is given by them.
I appreciate any hints that could shed light on this question.
All best,
Bphil
