Geometry solution involving complex numbers from USAMO

analytic geometrycomplex numberscontest-mathgeometry

Quadrilateral $AP BQ$ is inscribed in circle $ω$ with $∠P = ∠Q = 90^{\circ}$
and $AP = AQ < BP$. Let $X$ be a variable point on segment $P Q$. Line
$AX$ meets $ω$ again at $S$ (other than $A$). Point $T$ lies on arc
$AQB$ of $ω$ such that $XT$ is perpendicular to $AX$. Let $M$ denote
the midpoint of chord $ST$. As $X$ varies on segment $P Q$, show that
$M$ moves along a circle. (USAMO 2015/P2)

Okay so, I'm studying geometry from the book EGMO by Evan Chan and this was a practice problem. The solution in the back of the book is the same as the one from Evan's 2015 USAMO notes. I am pretty much a newbie with geometry with complex numbers.

Everything in his solution makes sense apart from this one part. Could somebody please explain that? Also was this question supposed to be trivial using complex geometry? (Just askin ).

The solution is as follows:

Toss on the complex unit circle with $a = −1$, $b = 1$, $z=-\frac{1}{2}$. Let $s$ and $t$ be on the unit circle. We claim $z$ is the center. It follows from standard formulas that $x =\frac{1}{2}(s
+ t − 1 +\frac{s}{t})$

thus,

$4 \cdot \mathrm{Re}(x) + 2 = s + t +\frac{1}{s}+\frac{1}{t}+\frac{t}{s}+\frac{s}{t}$

which depends only on $P $ and $Q$, and not on $X$. Thus,

$4 \left| z − \dfrac{s + t}{2}\right|^2= |s + t + 1|^2 = 3 + (4 \cdot \mathrm{Re}(x )+ 2)$

does not depend on $X$.

Well I guess I get that $\mathrm{Re}(x)$ refers to the real part of $x$ but where does the quantity $4\cdot \mathrm{Re}(x)+2$ come from?

Also in the next equation there is $4 \left|z − \dfrac{s + t}{2}\right|^2$. Where does this come from? And why does not being dependent on $X$ mean done? Please forgive my stupidity in case this is all extremely trivial stuff.
Thanks a lot.

Best Answer

The goal is to show that the locus of the midpoint of $ST$ is a circle. The equation $ 4 | z - \frac{ s + t } { 2} | ^2 = A $ would then mean that the midpoint of $ST$, represented by $ \frac{ s+ t } { 2 } $ lies on a circle of radius $ \sqrt{\frac{ A}{4}} $ about the point $z$. Hence, it suffices to show that $A$ is a constant (independent of the point $X$ chosen).

  1. To prove this, Evan claims that $ Z =( - \frac{1}{2},0) $ is the center. (You could guess this from the construction, or force it out based on the calculations below.) Setting $ z = - \frac{1}{2}$ , we get $ 4 | z - \frac{ s + t } { 2} | ^2 = |s + t + 1|^2$. (This is just arithmetic)
  2. Now, because $ s\overline{s} = 1$, hence $ \overline{s} = \frac{1}{s}$. Likewise, $ \overline{t} = \frac{1}{t} $. Hence, $ |s+t+1| ^2 = (s+t+1 ) ( \overline{s+t+1} ) = (s+t+1) ( \frac{1}{s} + \frac{1}{t} + 1 ) = 3 + \frac{s}{t} + \frac{t}{s} + s + t + \frac{1}{s} + \frac{1}{t}. $
  3. Substituting $4 Re x + 2 = s + t +\frac{1}{s}+\frac{1}{t}+\frac{t}{s}+\frac{s}{t}$ (verified below) into the above, we get $ | s+t+1|^2 = 4 Re x + 2 + 3 $.
  4. Notice that $PQ$ is a vertical line, and $X$ is a point on $PQ$, so the real part of $X$ is a constant.
  5. Putting all of this together, we do truly get $ 4 | z - \frac{x+t}{2} |^2 = |s+t+1|^2 = 4Rex + 5= A$, so the locus is a circle.

Using that $x = \frac{1}{2} ( s + t - 1 + \frac{s}{t})$, $ \overline{s} = \frac{1}{s}$, and $ \overline{t} = \frac{1}{t} $, hence $$ 4 Re x = 2(x + \overline{x}) = ( s + t - 1 + \frac{s}{t} ) + (\overline{ s + t - 1 + \frac{s}{t} } ) = s + t - 1 + \frac{s}{t} + \frac{1}{s} + \frac{1}{t} - 1 + \frac{t}{s} . $$

Note: I did not come up with $x = \frac{1}{2} ( s + t - 1 + \frac{s}{t})$ when I first stated on this. I only knew about it from your writeup. While I can prove it having known it, I'm not confident that I would have come up with it independently.

Yes, this question ends up being quite direct once we use complex numbers. The "hard" part was

  1. Guessing the center $z$ but even that wasn't that bad if you backtracked from the available information, and
  2. Finding the relationship $x = \frac{1}{2} ( s + t - 1 + \frac{s}{t})$

Part of the reason why complex numbers makes the solution trivial than other techniques is that the relationship $x = \frac{1}{2} ( s + t - 1 + \frac{s}{t})$ would be hard to express otherwise. E.g. If you saw it, how would you describe the relationship of these 3 points? Even using vectors (which is sometimes a close substitute for complex numbers), this is more complicated than most would like to deal with.

Related Question