1976 USAMO Problems/Problem 2

Problem

If $A$ and $B$ are fixed points on a given circle and $XY$ is a variable diameter of the same circle, determine the locus of the point of intersection of lines $AX$ and $BY$ . You may assume that $AB$ is not a diameter.

$[asy] size(300); defaultpen(fontsize(8)); real r=10; picture pica, picb; pair A=r*expi(5*pi/6), B=r*expi(pi/6), X=r*expi(pi/3), X1=r*expi(-pi/12), Y=r*expi(4*pi/3), Y1=r*expi(11*pi/12), O=(0,0), P, P1; P = extension(A,X,B,Y);P1 = extension(A,X1,B,Y1); path circ1 = Circle((0,0),r); draw(pica, circ1);draw(pica, B--A--P--Y--X);dot(pica,P^^O); label(pica,"$A$",A,(-1,1));label(pica,"$B$",B,(1,0));label(pica,"$X$",X,(0,1));label(pica,"$Y$",Y,(0,-1));label(pica,"$P$",P,(1,1));label(pica,"$O$",O,(-1,1));label(pica,"(a)",O+(0,-13),(0,0)); draw(picb, circ1);draw(picb, B--A--X1--Y1--B);dot(picb,P1^^O); label(picb,"$A$",A,(-1,1));label(picb,"$B$",B,(1,1));label(picb,"$X$",X1,(1,-1));label(picb,"$Y$",Y1,(-1,0));label(picb,"$P'$",P1,(-1,-1));label(picb,"$O$",O,(-1,-1)); label(picb,"(b)",O+(0,-13),(0,0)); add(pica); add(shift(30*right)*picb); [/asy]$

Solution

WLOG, assume that the circle is the unit circle centered at the origin. Then the points $A$ and $B$ have coordinates $(-a,b)$ and $(a,b)$ respectively and $X$ and $Y$ have coordinates $(r,s)$ and $(-r,-s)$ . Note that these coordinates satisfy $a^2 + b^2 = 1$ and $r^2 + s^2 = 1$ since these points are on a unit circle. Now we can find equations for the lines: $\begin{align*} AX \longrightarrow y &= \frac{(s-b)x+rb+sa}{r+a}\\ BY \longrightarrow y &= \frac{(s+b)x+rb-sa}{r+a}. \end{align*}$ Solving these simultaneous equations gives coordinates for $P$ in terms of $a, b, r,$ and $s$ : $P = \left(\frac{as}{b},\frac{1 - ar}{b}\right)$ . These coordinates can be parametrized in Cartesian variables as follows: $\begin{align*} x &= \frac{as}{b}\\ y &= \frac{1 - ar}{b}. \end{align*}$ Now solve for $r$ and $s$ to get $r = \frac{1-by}{a}$ and $s = \frac{bx}{a}$ . Then since $r^2 + s^2 = 1, \left(\frac{bx}{a}\right)^2 + \left(\frac{1-by}{a}\right)^2 = 1$ which reduces to $x^2 + (y-1/b)^2 = \frac{a^2}{b^2}.$ This equation defines a circle and is the locus of all intersection points $P$ . In order to define this locus more generally, find the slope of this circle function using implicit differentiation: $\begin{align*} 2x + 2(y-1/b)y' &= 0\\ (y-1/b)y' &= -x\\ y' &= \frac{-x}{y-1/b}. \end{align*}$ Now note that at points $A$ and $B$ , this slope expression reduces to $y' = \frac{-b}{a}$ and $y' = \frac{b}{a}$ respectively, values which are identical to the slopes of lines $AO$ and $BO$ . Thus we conclude that the complete locus of intersection points is the circle tangent to lines $AO$ and $BO$ at points $A$ and $B$ respectively.

Solution 2

Notice that $m\angle AYP=\frac{1}{2}m\overarc{AB}$ (Inscried angle theorem) and that $m\angle XAY=90$ since $XY$ is a diameter, and thus subtends an arc of $180$ . This will hold for all $X$ and all $Y$ , and so by AA similarity, the angle $APY$ will be constant for all P, thus implying that the points A, B, and all P will be concyclic. If we assume that the center of this circle is $H$ , we know that $m\angle AHB=2m\angle APB=2(90-\frac{1}{2}\overarc{AB})=180-\overarc{AB}$ . We can assume that $\overline{AH}$ and $\overline{BH}$ intersect the original circle at points $M$ , and $N$ respectively. This will give us that $m\angle AHB=\frac{1}{2}|m\overarc{MN}-m\overarc{AB}|$ (Since $H$ lies on the perpendicular bisector of $AB$ , we know that $M$ , $N$ will be on the same side of $AB$ .) Now we also know that $m\overarc{MN}\leq m\overarc{AB}$ or $m\overarc{MN}\leq 360-m\overarc{AB}$ . The only case where $m\overarc{MN}$ satisfies the measure of $\angle AHB$ , is when $M=A$ and $N=B$ , implying that $AH$ and $BH$ are tangents, and so $H$ is the intersection of the tangents from $A$ and $B$ to the original circle.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

1976 USAMO (Problems • Resources)
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1976 USAMO Problems/Problem 2

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Solution 2

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