Difference between revisions of "2005 USAMO Problems/Problem 3"

Latest revision as of 00:17, 11 October 2021

Problem

(Zuming Feng) Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on side $BC$ . Construct point $C_1$ in such a way that convex quadrilateral $APBC_1$ is cyclic, $QC_1 \parallel CA$ , and $C_1$ and $Q$ lie on opposite sides of line $AB$ . Construct point $B_1$ in such a way that convex quadrilateral $APCB_1$ is cyclic, $QB_1 \parallel BA$ , and $B_1$ and $Q$ lie on opposite sides of line $AC$ . Prove that points $B_1, C_1,P$ , and $Q$ lie on a circle.

Solution

Solution 1

Let $B_1'$ be the second intersection of the line $C_1A$ with the circumcircle of $APC$ , and let $Q'$ be the second intersection of the circumcircle of $B_1' C_1P$ and line $BC$ . It is enough to show that $B_1'=B_1$ and $Q' =Q$ . All our angles will be directed, and measured mod $\pi$ .

$[asy] size(300); defaultpen(1); pair A=(2,5), B=(-1,0), C=(5,0); pair C1=(.5,5.7); path O1=circumcircle(A,B,C1); pair P=IntersectionPoint(O1,B--C,1); path O2=circumcircle(A,P,C); pair B1=IntersectionPoint(O2,C1--5A-4C1,0); path O=circumcircle(B1,C1,P); pair Q=IntersectionPoint(O,B--C,1); draw(C1--P--A--B--C--A); draw(P--B1--C1--Q--B1); draw(O1,dashed+linewidth(.7)); draw(O2,dashed+linewidth(.7)); draw(O,dotted); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$P$",P,S); label("$Q'$",Q,S); label("$C_1$",C1,N); label("$B_1'$",B1,E); [/asy]$

Since points $C_1, P, Q', B_1'$ are concyclic and points $C_1, A,B_1'$ are collinear, it follows that $\angle C_1 Q' P \equiv \angle C_1 B_1' P \equiv \angle A B_1' P .$ But since points $A, B_1', P, C$ are concyclic, $\angle AB_1'P \equiv \angle ACP .$ It follows that lines $AC$ and $C_1 Q'$ are parallel. If we exchange $C$ with $B$ and $C_1$ with $B_1'$ in this argument, we see that lines $AB$ and $B_1' Q'$ are likewise parallel.

It follows that $Q'$ is the intersection of $BC$ and the line parallel to $AC$ and passing through $C_1$ . Hence $Q' = Q$ . Then $B_1'$ is the second intersection of the circumcircle of $APC$ and the line parallel to $AB$ passing through $Q$ . Hence $B_1' = B_1$ , as desired. $\blacksquare$

Motivation: One can notice that if you take such a $B_1'$ then $PQB_1'C_1$ is cyclic, and that similarly $PQB_1C_1'$ is also cyclic. One gets the intuition that only one such circle should exist where the other chord passes through A, and so sets up a ghost point for $Q$ , which works. ~cocohearts

Solution 2

Lemma. $B_1, A, C_1$ are collinear.

Suppose they are not collinear. Let line $B_1 A$ intersect circle $ABP$ (i.e. the circumcircle of $ABP$ ) again at $C_2$ distinct from $C_1$ . Because $\angle C_2 B_1 Q = \angle C_2 AB = \angle C_2 PB = 180^\circ - \angle C_2 PQ$ , we have that $B_1 C_2 PQ$ is cyclic. Hence $\angle C_2 QP = \angle C_2 B_1 P = \angle P B_1 A = \angle C$ , so $C_2 Q // AC$ . Then $C_2$ must be the other intersection of the parallel to $AC$ through $Q$ with circle $ABP$ . Then $C_2$ is on segment $C_1 Q$ , so $C_2$ is contained in triangle $ABQ$ . However, line $AB_1$ intersects this triangle only at point $A$ because $B_1$ lies on arc $AC$ not containing $P$ of circle $APC$ , a contradiction. Hence, $B_1, A, C_1$ are collinear, as desired.

As a result, we have $\angle C_1 B_1 Q = \angle C_1 AB = \angle C_1 PB = 180^\circ - \angle C_1 PQ$ , so $B_1 C_1 PQ$ is cyclic, as desired.

Solution 3

Due to the parallel lines, $m\angle C_1QB_1=m\angle A.$ Therefore, it suffices to prove that $m\angle C_1PB_1=m\angle C_1BA+m\angle ACB_1=m\angle A.$ Note that $m\angle BC_1A+m\angle AB_1C=180^{\circ}$ by the cyclic quadrilaterals. Now, the condition simplifies to proving $C_1,A,B_1$ collinear.

Use barycentric coordinates. Let $A=(1,0,0), B=(0,1,0), C=(0,0,1), P=(0,p,1-p), Q=(0,q,1-q).$ By the parallel lines, $C_1=(1-q-z',q,z'), B_1=(q-y',y',1-q)$ for some $y',z'.$ As $A$ is a vertex of the reference triangle, we must prove that $q(1-q)=y'z'.$

Now we find the circumcircle of $\triangle APC.$ Let its equation be $-a^2yz-b^2xz-c^2xy+(x+y+z)(ux+vy+wz)=0.$ Substituting in $A,C$ gives $u=w=0.$ Substituting in $P$ yields $-a^2p(1-p)+vp=0\implies v = a^2(1-p).$ Substituting in $B_1$ yields $0=-a^2y'(1-q)-b^2(q-y')(1-q)-c^2y'(q-y')+a^2(1-p)y'=0.$ $c^2y'^2+(a^2q-a^2p+b^2(1-q)-c^2q)y'-b^2q(1-q)=0$

Similarly, the circumcircle of $\triangle APB$ is $-a^2yz-b^2xz-c^2xy+(x+y+z)a^2pz.$ Now, we substitute $C_1$ into the equation

$0=-a^2qz'-b^2z'(1-q-z')-c^2q(1-q-z')+a^2pz'=0$ $b^2 z'^2+(a^2p-a^2q-b^2(1-q)+c^2q)z'-c^2q(1-q)=0$

Let $k=a^2p-a^2q-b^2(1-q)+c^2q,$ and note that both $y',z'$ are negative. By the quadratic formula, we have $y'=\frac{k-\sqrt{k^2+4b^2c^2q(1-q)}}{2c^2}, z'=\frac{-k-\sqrt{k^2+4b^2c^2q(1-q)}}{2b^2}$ Multiplying these two, we have $y'z'=\frac{4b^2c^2q(1-q)}{4b^2c^2}=q(1-q),$ as desired. $\blacksquare$

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

@@ Line 3: / Line 3: @@
 == Solution ==
+=== Solution 1 ===
 Let <math>B_1'</math> be the second intersection of the line <math>C_1A</math> with the circumcircle of <math>APC</math>, and let <math>Q'</math> be the second intersection of the circumcircle of <math>B_1' C_1P</math> and line <math>BC</math>.  It is enough to show that <math>B_1'=B_1</math> and <math>Q' =Q</math>.  All our angles will be directed, and measured mod <math>\pi</math>.
@@ Line 40: / Line 42: @@
 It follows that <math>Q'</math> is the intersection of <math>BC</math> and the line parallel to <math>AC</math> and passing through <math>C_1</math>.  Hence <math>Q' = Q</math>.  Then <math>B_1'</math> is the second intersection of the circumcircle of <math>APC</math> and the line parallel to <math>AB</math> passing through <math>Q</math>.  Hence <math>B_1' = B_1</math>, as desired.  <math>\blacksquare</math>
+Motivation: One can notice that if you take such a <math>B_1'</math> then <math>PQB_1'C_1</math> is cyclic, and that similarly <math>PQB_1C_1'</math> is also cyclic. One gets the intuition that only one such circle should exist where the other chord passes through A, and so sets up a ghost point for <math>Q</math>, which works. ~cocohearts
+===Solution 2===
+''Lemma''. <math>B_1, A, C_1</math> are collinear.
+Suppose they are not collinear. Let line <math>B_1 A</math> intersect circle <math>ABP</math> (i.e. the circumcircle of <math>ABP</math>) again at <math>C_2</math> distinct from <math>C_1</math>. Because <math>\angle C_2 B_1 Q = \angle C_2 AB = \angle C_2 PB = 180^\circ - \angle C_2 PQ</math>, we have that <math>B_1 C_2 PQ</math> is cyclic. Hence <math>\angle C_2 QP = \angle C_2 B_1 P = \angle P B_1 A = \angle C</math>, so <math>C_2 Q // AC</math>. Then <math>C_2</math> must be the other intersection of the parallel to <math>AC</math> through <math>Q</math> with circle <math>ABP</math>. Then <math>C_2</math> is on segment <math>C_1 Q</math>, so <math>C_2</math> is contained in triangle <math>ABQ</math>. However, line <math>AB_1</math> intersects this triangle only at point <math>A</math> because <math>B_1</math> lies on arc <math>AC</math> not containing <math>P</math> of circle <math>APC</math>, a contradiction. Hence, <math>B_1, A, C_1</math> are collinear, as desired.
+As a result, we have <math>\angle C_1 B_1 Q = \angle C_1 AB = \angle C_1 PB = 180^\circ - \angle C_1 PQ</math>, so <math>B_1 C_1 PQ</math> is cyclic, as desired.
+=== Solution 3 ===
+Due to the parallel lines, <math>m\angle C_1QB_1=m\angle A.</math> Therefore, it suffices to prove that <cmath>m\angle C_1PB_1=m\angle C_1BA+m\angle ACB_1=m\angle A.</cmath> Note that <math>m\angle BC_1A+m\angle AB_1C=180^{\circ}</math> by the cyclic quadrilaterals. Now, the condition simplifies to proving <math>C_1,A,B_1</math> collinear.
+Use barycentric coordinates. Let <cmath>A=(1,0,0), B=(0,1,0), C=(0,0,1), P=(0,p,1-p), Q=(0,q,1-q).</cmath> By the parallel lines, <cmath>C_1=(1-q-z',q,z'), B_1=(q-y',y',1-q)</cmath> for some <math>y',z'.</math> As <math>A</math> is a vertex of the reference triangle, we must prove that <math>q(1-q)=y'z'.</math>
+Now we find the circumcircle of <math>\triangle APC.</math> Let its equation be <cmath>-a^2yz-b^2xz-c^2xy+(x+y+z)(ux+vy+wz)=0.</cmath> Substituting in <math>A,C</math> gives <math>u=w=0.</math> Substituting in <math>P</math> yields <cmath>-a^2p(1-p)+vp=0\implies v = a^2(1-p).</cmath> Substituting in <math>B_1</math> yields
+<cmath>0=-a^2y'(1-q)-b^2(q-y')(1-q)-c^2y'(q-y')+a^2(1-p)y'=0.</cmath>
+<cmath>c^2y'^2+(a^2q-a^2p+b^2(1-q)-c^2q)y'-b^2q(1-q)=0</cmath>
+Similarly, the circumcircle of <math>\triangle APB</math> is <cmath>-a^2yz-b^2xz-c^2xy+(x+y+z)a^2pz.</cmath> Now, we substitute <math>C_1</math> into the equation
+<cmath>0=-a^2qz'-b^2z'(1-q-z')-c^2q(1-q-z')+a^2pz'=0</cmath>
+<cmath>b^2 z'^2+(a^2p-a^2q-b^2(1-q)+c^2q)z'-c^2q(1-q)=0</cmath>
+Let <math>k=a^2p-a^2q-b^2(1-q)+c^2q,</math> and note that both <math>y',z'</math> are negative. By the quadratic formula, we have
+<cmath>y'=\frac{k-\sqrt{k^2+4b^2c^2q(1-q)}}{2c^2}, z'=\frac{-k-\sqrt{k^2+4b^2c^2q(1-q)}}{2b^2}</cmath>
+Multiplying these two, we have <cmath>y'z'=\frac{4b^2c^2q(1-q)}{4b^2c^2}=q(1-q),</cmath> as desired. <math>\blacksquare</math>
 {{alternate solutions}}

2005 USAMO (Problems • Resources)
Preceded by Problem 2	Followed by Problem 4
1 • 2 • 3 • 4 • 5 • 6
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