Difference between revisions of "2003 USAMO Problems/Problem 4"

Latest revision as of 01:39, 10 January 2023

Problem

Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$ , respectively. Lines $AB$ and $DE$ intersect at $F$ , while lines $BD$ and $CF$ intersect at $M$ . Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$ .

Solutions

Solution 1

Extend segment $DM$ through $M$ to $G$ such that $FG\parallel CD$ . $[asy] defaultpen(fontsize(10)+0.6); size(250); var theta=22, r=0.58; pair B=origin, A=dir(theta), C=A+(rotate(78)*0.8*A), O=IP(CR(B,r),CR(A,r)); path c=CR(O,r); pair D=IP(c,A--C), E=IP(c,B--C), F=extension(A,B,D,E), M=extension(B,D,C,F), G=extension(D,M,F,F+C-D); draw(A--B--C--A^^E--F--C^^A--F^^B--M^^E--M); draw(c); draw(F--G--C^^M--G,gray+0.4); dot("$A$",A,dir(F-E)); dot("$B$",B,2*dir(B-A)); dot("$C$",C,1.5*dir(C-A)); dot("$D$",D,2.5*dir(250)); dot("$E$",E,2.5*dir(C-A)); dot("$F$",F,dir(F-E)); dot("$M$",M,2.5*dir(255)); dot("$G$",G,dir(G-M)); [/asy]$ Then $MF = MC$ if and only if quadrilateral $CDFG$ is a parallelogram, or, $FD\parallel CG$ . Hence $MC = MF$ if and only if $\angle GCD = \angle FDA$ , that is, $\angle FDA + \angle CGF = 180^\circ$ .

Because quadrilateral $ABED$ is cyclic, $\angle FDA = \angle ABE$ . It follows that $MC = MF$ if and only if $180^\circ = \angle FDA + \angle CGF = \angle ABE + \angle CGF,$ that is, quadrilateral $CBFG$ is cyclic, which is equivalent to $\angle CBM = \angle CBG = \angle CFG = \angle DCF = \angle DCM.$ Because $\angle DMC = \angle CMB$ , $\angle CBM = \angle DCM$ if and only if triangles $BCM$ and $CDM$ are similar, that is $\frac{CM}{BM} = \frac{DM}{CM},$ or $MB\cdot MD = MC^2$ .

Solution 2

We first assume that $MB\cdot MD = MC^2$ . Because $\frac{MC}{MD} = \frac{MB}{MC}$ and $\angle CMD = \angle BMC$ , triangles $CMD$ and $BMC$ are similar. Consequently, $\angle MCD = \angle MBC$ . This exact condition can also be visualized through power of M with respect to the circumcircle of triangle $BDC$ and getting the angle condition from the alternate segment theorem. $[asy] defaultpen(fontsize(10)+0.6); size(250); var theta=22, r=0.58; pair B=origin, A=dir(theta), C=A+(rotate(78)*0.8*A), O=IP(CR(B,r),CR(A,r)); path c=CR(O,r); pair D=IP(c,A--C), E=IP(c,B--C), F=extension(A,B,D,E), M=extension(B,D,C,F), G=extension(D,M,F,F+C-D); draw(A--B--C--A^^E--F--C^^A--F^^B--M^^E--M); draw(c); draw(A--E,gray+0.4); dot("$A$",A,dir(F-E)); dot("$B$",B,2*dir(B-A)); dot("$C$",C,1.5*dir(C-A)); dot("$D$",D,2.5*dir(250)); dot("$E$",E,2.5*dir(C-A)); dot("$F$",F,dir(F-E)); dot("$M$",M,2.5*dir(255)); [/asy]$ Because quadrilateral $ABED$ is cyclic, $\angle DAE = \angle DBE$ . Hence $\angle FCA = \angle MCD = \angle MBC = \angle DBE = \angle DAE = \angle CAE,$ implying that $AE\parallel CF$ , so $\angle AEF = \angle CFE$ . Because quadrilateral $ABED$ is cyclic, $\angle ABD = \angle AED$ . Hence $\angle FBM = \angle ABD = \angle AED = \angle AEF = \angle CFE = \angle MFD.$ Because $\angle FBM = \angle DFM$ and $\angle FMB = \angle DMF$ , triangles $BFM$ and $FDM$ are similar. Consequently, $\frac{FM}{DM} = \frac{BM}{FM}$ , or $FM^2 = BM\cdot DM = CM^2$ . Therefore $MC^2 = MB\cdot MD$ implies $MC = MF$ .

Now we assume that $MC = MF$ . Applying Ceva's Theorem to triangle $BCF$ and cevians $BM, CA, FE$ gives $\frac{BA}{AF}\cdot\frac{FM}{MC}\cdot\frac{CE}{EB} = 1,$ implying that $\frac{BA}{AF} = \frac{BE}{EC}$ , so $AE\parallel CF$ .

Consequently, $\angle DCM = \angle DAE$ . Because quadrilateral $ABED$ is cyclic, $\angle DAE = \angle DBE$ . Hence $\angle DCM = \angle DAE = \angle DBE = \angle CBM.$ Because $\angle CBM = \angle DCM$ and $\angle CMB = \angle DMC$ , triangles $BCM$ and $CDM$ are similar. Consequently, $\frac{CM}{DM} = \frac{BM}{CM}$ , or $CM^2 = BM\cdot DM$ .

Combining the above, we conclude that $MF = MC$ if and only if $MB\cdot MD = MC^2$ .

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: @@
 Let <math>ABC</math> be a triangle. A circle passing through <math>A</math> and <math>B</math> intersects segments <math>AC</math> and <math>BC</math> at <math>D</math> and <math>E</math>, respectively. Lines <math>AB</math> and <math>DE</math> intersect at <math>F</math>, while lines <math>BD</math> and <math>CF</math> intersect at <math>M</math>. Prove that <math>MF = MC</math> if and only if <math>MB\cdot MD = MC^2</math>.
-== Solution ==
+== Solutions ==
-by April
+=== Solution 1 ===
-Take <math>G\in BD: \,FG\parallel CD</math>. We have:
+Extend segment <math>DM</math> through <math>M</math> to <math>G</math> such that <math>FG\parallel CD</math>.
+<asy>
+defaultpen(fontsize(10)+0.6); size(250);
+var theta=22, r=0.58;
+pair B=origin, A=dir(theta), C=A+(rotate(78)*0.8*A), O=IP(CR(B,r),CR(A,r));
+path c=CR(O,r);
+pair D=IP(c,A--C), E=IP(c,B--C), F=extension(A,B,D,E), M=extension(B,D,C,F), G=extension(D,M,F,F+C-D);
+draw(A--B--C--A^^E--F--C^^A--F^^B--M^^E--M); draw(c); draw(F--G--C^^M--G,gray+0.4);
+dot("$A$",A,dir(F-E)); dot("$B$",B,2*dir(B-A)); dot("$C$",C,1.5*dir(C-A)); dot("$D$",D,2.5*dir(250)); dot("$E$",E,2.5*dir(C-A)); dot("$F$",F,dir(F-E)); dot("$M$",M,2.5*dir(255)); dot("$G$",G,dir(G-M));
+</asy>
+Then <math>MF = MC</math> if and only if quadrilateral <math>CDFG</math> is a parallelogram, or, <math>FD\parallel CG</math>. Hence <math>MC = MF</math> if and only if <math>\angle GCD = \angle FDA</math>, that is, <math>\angle FDA + \angle CGF = 180^\circ</math>.
-<math>MF = MC\Longleftrightarrow \textrm{the quadrilateral}\; CDFG\; \textrm{is a parallelogram} \\
+Because quadrilateral <math>ABED</math> is cyclic, <math>\angle FDA = \angle ABE</math>. It follows that <math>MC = MF</math> if and only if
-\Longleftrightarrow FD\parallel CG\Longleftrightarrow\angle FDA = \angle GCD\Longleftrightarrow\angle FDA + \angle CGF = 180^\circ \\
+<cmath>180^\circ = \angle FDA + \angle CGF = \angle ABE + \angle CGF,</cmath>
-\Longleftrightarrow \angle ABE + \angle CGF = 180^\circ\Longleftrightarrow\textrm{the quadrilateral}\;CBGF\;\textrm{is cyclic} \\
+that is, quadrilateral <math>CBFG</math> is cyclic, which is equivalent to
-\Longleftrightarrow\angle CBM = \angle CBG = \angle CFG = \angle DCF = \angle DCM \\
+<cmath>\angle CBM = \angle CBG = \angle CFG = \angle DCF = \angle DCM.</cmath>
-\Longleftrightarrow\triangle BCM\sim\triangle CDM\Longleftrightarrow MB\cdot MD = MC^{2}</math>
+Because <math>\angle DMC = \angle CMB</math>, <math>\angle CBM = \angle DCM</math> if and only if triangles <math>BCM</math> and <math>CDM</math> are similar, that is
+<cmath>\frac{CM}{BM} = \frac{DM}{CM},</cmath>
+or <math>MB\cdot MD = MC^2</math>.
-== Resources ==
+=== Solution 2 ===
+We first assume that <math>MB\cdot MD = MC^2</math>. Because <math>\frac{MC}{MD} = \frac{MB}{MC}</math> and <math>\angle CMD = \angle BMC</math>, triangles <math>CMD</math> and <math>BMC</math> are similar. Consequently, <math>\angle MCD = \angle MBC</math>. This exact condition can also be visualized through power of M with respect to the circumcircle of triangle <math>BDC</math> and getting the angle condition from the alternate segment theorem.
+<asy>
+defaultpen(fontsize(10)+0.6); size(250);
+var theta=22, r=0.58;
+pair B=origin, A=dir(theta), C=A+(rotate(78)*0.8*A), O=IP(CR(B,r),CR(A,r));
+path c=CR(O,r);
+pair D=IP(c,A--C), E=IP(c,B--C), F=extension(A,B,D,E), M=extension(B,D,C,F), G=extension(D,M,F,F+C-D);
+draw(A--B--C--A^^E--F--C^^A--F^^B--M^^E--M); draw(c); draw(A--E,gray+0.4);
+dot("$A$",A,dir(F-E)); dot("$B$",B,2*dir(B-A)); dot("$C$",C,1.5*dir(C-A)); dot("$D$",D,2.5*dir(250)); dot("$E$",E,2.5*dir(C-A)); dot("$F$",F,dir(F-E)); dot("$M$",M,2.5*dir(255));
+</asy>
+Because quadrilateral <math>ABED</math> is cyclic, <math>\angle DAE = \angle DBE</math>. Hence
+<cmath>\angle FCA = \angle MCD = \angle MBC = \angle DBE = \angle DAE = \angle CAE,</cmath>
+implying that <math>AE\parallel CF</math>, so <math>\angle AEF = \angle CFE</math>. Because quadrilateral <math>ABED</math> is cyclic, <math>\angle ABD = \angle AED</math>. Hence
+<cmath>\angle FBM = \angle ABD = \angle AED = \angle AEF = \angle CFE = \angle MFD.</cmath>
+Because <math>\angle FBM = \angle DFM</math> and <math>\angle FMB = \angle DMF</math>, triangles <math>BFM</math> and <math>FDM</math> are similar. Consequently, <math>\frac{FM}{DM} = \frac{BM}{FM}</math>, or <math>FM^2 = BM\cdot DM = CM^2</math>. Therefore <math>MC^2 = MB\cdot MD</math> implies <math>MC = MF</math>.
-* [[2003 USAMO Problems]]
+Now we assume that <math>MC = MF</math>. Applying Ceva's Theorem to triangle <math>BCF</math> and cevians <math>BM, CA, FE</math> gives
-* [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=53670 Discussion on AOPS/MathLinks]
+<cmath>\frac{BA}{AF}\cdot\frac{FM}{MC}\cdot\frac{CE}{EB} = 1,</cmath>
+implying that <math>\frac{BA}{AF} = \frac{BE}{EC}</math>, so <math>AE\parallel CF</math>.
+Consequently, <math>\angle DCM = \angle DAE</math>. Because quadrilateral <math>ABED</math> is cyclic, <math>\angle DAE = \angle DBE</math>. Hence
+<cmath>\angle DCM = \angle DAE = \angle DBE = \angle CBM.</cmath>
+Because <math>\angle CBM = \angle DCM</math> and <math>\angle CMB = \angle DMC</math>, triangles <math>BCM</math> and <math>CDM</math> are similar. Consequently, <math>\frac{CM}{DM} = \frac{BM}{CM}</math>, or <math>CM^2 = BM\cdot DM</math>.
+Combining the above, we conclude that <math>MF = MC</math> if and only if <math>MB\cdot MD = MC^2</math>.
+{{alternate solutions}}
+== See also ==
+{{USAMO newbox|year=2003|num-b=3|num-a=5}}
 [[Category:Olympiad Geometry Problems]]
+{{MAA Notice}}

2003 USAMO (Problems • Resources)
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