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Method B3

Here we extend the animation by letting B" and C" move beyond point A. We also show the circle through B, C, B", C".
Note that we can still say AB''.AE = AC".AG, ie AB".AC = AC".AB or, using the notation on an earlier page, AD.AC = AE.AB.

When B" and C" move beyond point A, we are back to Elements Book 2, Proposition 35 about intersecting chords.

Finally we look at a converse of our original task. We can argue that is possible to draw a circle throught the 'base vertices' of similar triangles like ABC and ADE (ie through B, C, D and E). Can we do the same for the 'base vertices' of similar triangles like ADE and AFG (ie through E, D, F and G)? We consider this here: PS