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\title{TJUSAMO Practice 14 - Geometry}
\author{PDiao06}
\date{\today}

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\maketitle

Thankfully, according to Thomas Mildorf, geometry is low-tech on the USAMO.  This means that most of those really crazy
things you always hear about are not worth your time learning at this point if you don't already know them.  Mainly, the
big topics are similar triangles, cyclic quads, and power of a point.  Other geometry that you have learned from short
problems are probably sufficient to make most geometry at least accessible to you.  Before even starting, I'll give you
guys some tips I've received from people much better at geometry than I.

\section{Tips From Some Masters}

\subsection{Master 1 - Zuming Feng aka the Zumbot}

I'm pretty sure he didn't say these things, but they sound reasonable to me.

\begin{itemize}

\item Draw good diagrams.  This means big ones, accurate ones, and bring your compass, ruler, protractor, graph paper, take
advantage of this stuff.
\item Make cyclic quads and parallel lines.  They are handy for everything!
\item Don't forget trig.
\item Consider the area.
\item Find the special point of concurrence.

\end{itemize}

\subsection{Master 2 - Richard Rucccccyzzks}

I think I mispelled that.  It's a reflection of the fact that I can't pronounce that name.  These are actually directly from
him.

\begin{itemize}

\item Big diagrams (he says it too!  Pictures tell you the answers that you have to prove.  They tell you intermediate 
steps and also help you work backwards.  It's the equivalent of trying small numbers in a number theory problem, it quickly
tells you an angle is a right angle if every time you draw it, it looks like one!)
\item Don't panic. (I don't know how that's gonna help.)
\item Work in two directions. (absolutely crucial)
\item Ask yourself what you haven't used yet when stuck.

\end{itemize}

\section{Basic tool development}

\subsection{Similar triangles}

There isn't much to say about them.  Angle's the same and side ratios.  It is surprising how darn useful they are and how
noticing a clever similar triangle often finishes a problem.  For fun, prove angle bisector theorem!  

\subsection{Circle stuff}

\begin{enumerate}

\item Cyclic Quadrilaterals - go back to an earlier TJUSAMO and see the lecture on these.  They are crucial for angle chasing
and are very useful in other ways too such as Ptolemy's.  
\item Angles - prove the intercepted angles stuff to yourself. All of them, no need to memorize, just prove it to yourself.  
\item Power of Point? What, who needs that!  Prove it and then promptly forget such a useless piece of junk existed.  The
proof gives you far more information than the theorem itself.
\item Radical Axis Theorem - I was just kidding about the power of a point thing, it is actually extremely useful.  This 
theorem you might not know yet.  The radical axis of two circles is the locus of points that has equal powers with respect
to two different circles.  It is always a line (this is proved by ugly coordinates).  The radical axis theorem states that
given three circles, the three radical axes always concur at one point or are parallel.

\end{enumerate}

\section{Practice}

Here are some fun problems we can do together or you can do alone after this practice.

\begin{enumerate}

\item (USAMO 2003 Number 3) ABC is a triangle. A circle through A and B meets the sides AC, BC at D, E respectively. The 
lines AB and DE meet at F. The lines BD and CF meet at M. Show that M is the midpoint of CF iff $MB*MD = MC^2$.

\item (WOOT) WXYZ is a cyclic quadrilateral with diagonals that meet at Q. M is the midpoint of YZ. The circle through Q 
which is tangent to YZ at M meets WY at T and XZ at P. R is on XQ such that XR = ZP. S is on WQ such that $SR || WX$. Prove 
that WS = TY.

\item (WOOT) Line AB is tangent to circle O at point Y, with Y between A and B on the line. Point X is on circle O such 
that XY is a diameter. XA and XB meet the circle again at C and D, respectively. AD and BC meet the circle again at 
E and F, respectively. Prove that XE = XF.

\item Let $\omega$ be the circumcircle of triangle ABC, with orthocenter H. Let altitude AD intersect $\omega$ at A'. 
Prove that $HD = DA'$.

\item Three congruent circles pass through a common point P. The other points of intersection are A, B, and C.
Prove that P is the orthocenter of triangle ABC and prove that the circumcircle of triangle ABC has the same 
radius as the three original circles.

\item (WOOT) A, B, C, D are circles such that A and B touch externally at P, B and C touch externally at Q, C and D touch 
externally at R, and D and A touch externally at S. A does not intersect C, and B does not intersect D. Show that PQRS is 
cyclic.

\item (IMO Shortlist 2004 G1) Let ABC be an acute-angled triangle with $AB \neq AC$.  The circle with diameter BC intersects
the sides AB and AC at M and N, respectively.  Denote by O the midpoint of BC.  The bisectors of the angles BAC and MON 
intersect at R.  Prove that the circumcircles of the triangles BMR and CNR have a common point lying on the line segment BC.

\item (MOSP 2005) Let ABC be a triangle.  Suppose that the circle through C tangent to AB at A and the cricle through B 
tangent to AC at A have different radii, and let D be their second intersection.  Let E be the point on the ray AB such that
$AB = BE$.  Let F be the second intersection of the ray CA with the cricle through A,D,E.  Prove that $AF = AC$.

\item (MOSP 2005) Line AB is tangent to Circle $\omega_1$ at B.  Let C be a point not on $\omega_1$ such that segment AC 
meets $\omega_1$ at two distinct points.  Circle $\omega_2$ is tangent to line AC and $\omega_1$ at C and D, respectively,
such that D and B are on opposite sides of line AC.  Prove that the circumcenter of triangle BCD lies on the circumcircle
of triangle ABC.

\end{enumerate}

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