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 Infinite triangles share circumcircle and inscribed circle
01-22-2013, 10:03 PM
Post: #1
 elim Moderator Posts: 578 Joined: Feb 2010 Reputation: 0
Infinite triangles share circumcircle and inscribed circle
Let $\Delta ABC$ and so its circumcircle and inscribed circle $\odot O,\; \odot P$ be given,
and let $T \in \odot P,\quad F,\, F \in \odot O$ with $\overline{EF}$ being tangent to $\odot P$ at $T$.
Take $D\in \odot O$ such that $\overline{PF}$ bisects $\angle EFD$.

Prove that $DE$ is tangent to $\odot P$ hence $\odot P$ is the inscribed circle of $\Delta DEF$

01-22-2013, 10:23 PM
Post: #2
 elim Moderator Posts: 578 Joined: Feb 2010 Reputation: 0
RE: Infinite triangles share circumcircle and inscribed circle
Found this by accident, realized that the general case was already solved after some googling.

http://en.wikipedia.org/wiki/Poncelet%27s_porism

In geometry, Poncelet's porism (sometimes referred to as Poncelet's closure theorem), named after French engineer and mathematician Jean-Victor Poncelet, states the following: Let C and D be two plane conics. If it is possible to find, for a given n > 2, one n-sided polygon that is simultaneously inscribed in C and circumscribed around D, then it is possible to find infinitely many of them.

http://mathworld.wolfram.com/PonceletsPorism.html

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