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We have a triangle ABC with a circumscribed circle. Somewhere between BC we place a point D. There is a circle which goes through D and whose tangent at AB is A. This circle also intersects the circumscribed circle of ABC at a point E. Construct it.

So we're just going to construct a circle through points A and D, then see where it intersects ABC. So I originally thought that every point on the bisector of AD would work, but apparently not.

The answer sheet says that the center of the circle is the intersection of the bisector of AD and the line perpendicular to AB through the point A.

Why the perp line through A? I don't understand..

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  • $\begingroup$ How is the circumscribed circle of $ABC$ relevant to the problem? $\endgroup$ – Henning Makholm Nov 14 '13 at 16:25
  • $\begingroup$ @HenningMakholm Point E lies on it, not on ABC, that was my mistake.. $\endgroup$ – Phaptitude Nov 14 '13 at 16:34
  • $\begingroup$ It seems to be they want the circle to intersect AB at the point A only. $\endgroup$ – OR. Nov 14 '13 at 16:40
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It doesn't look like you have enough information to determine $E$ from what you have written. Since any three non-collinear points determine a circle, you could select $E$ anywhere on the circumscribed circle (except for $A$ and the other end of the diameter through $A$ and $D$) and then there'd be a circle through $A$, $D$ and $E$.

The answer sheet says that it is the intersection of the bisector of AD and the line perpendicular to AB through the point A.

That point generally won't like on the circumscribed circle at all. It is, however, the center of the circle that goes through $D$ and whose tangent at $A$ is $AB$.

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  • $\begingroup$ I'm so sorry, I incorrectly explained a crucial detail, please check the Q again. $\endgroup$ – Phaptitude Nov 14 '13 at 16:33
  • $\begingroup$ Is the question still unanswerable? $\endgroup$ – Phaptitude Nov 14 '13 at 16:38
  • $\begingroup$ @Phaptitude: As written, yes. Are you sure you didn't mean "tangent to $AB$ at $A$" instead of "intersects $AB$ at $A$"? $\endgroup$ – Henning Makholm Nov 14 '13 at 16:40
  • $\begingroup$ I'm extremely discombobulated, but yes, you are right, that's how they say you can find the center of the circle. And yes, you are right, tangent, not intersects. I was confused, because I thought for a point it means the same $\endgroup$ – Phaptitude Nov 14 '13 at 16:42
  • $\begingroup$ @Phaptitude: When we say that two curves "touch" at a point $P$, what we generally mean is that $P$ is on both curves, but there's a neighborhood of $P$ such that the two curves are on the same side of each other in the entire neighborhood except $P$. A circle and its tangent is kind of a prototypical example of this, but the curves passing through each other at an angle doesn't count as "touching". "Intersecting" may mean either "has a point in common" or "has a point in common without touching", depending on the context. $\endgroup$ – Henning Makholm Nov 14 '13 at 16:47

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