2003 USAMO Problems/Problem 4

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 $MB\cdot MD = MC^2$.

Solution

by April

Take $G\in BD: \,FG\parallel CD$. We have:

$MF = MC\Longleftrightarrow \textrm{the quadrilateral}\; CDFG\; \textrm{is a parallelogram} \\ \Longleftrightarrow FD\parallel CG\Longleftrightarrow\angle FDA = \angle GCD\Longleftrightarrow\angle FDA + \angle CGF = 180^\circ \\ \Longleftrightarrow \angle ABE + \angle CGF = 180^\circ\Longleftrightarrow\textrm{the quadrilateral}\;CBGF\;\textrm{is cyclic} \\ \Longleftrightarrow\angle CBM = \angle CBG = \angle CFG = \angle DCF = \angle DCM \\ \Longleftrightarrow\triangle BCM\sim\triangle CDM\Longleftrightarrow MB\cdot MD = MC^{2}$

Added diagram:

[asy] import graph; size(5cm);  real labelscalefactor = 0.5;  pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);   pen dotstyle = black;   real xmin = -4.3, xmax = 6.36, ymin = -3.98, ymax = 6.3;   draw((-0.16,3.1)--(-2.48,0.52));  draw((-2.48,0.52)--(3.78,0.52));  draw((3.78,0.52)--(-0.16,3.1));  draw(circle((-0.42,1), 2.12));  draw((1.04,4.43)--(3.78,0.52));  draw((1.04,4.43)--(-2.48,0.52));  draw((1.04,4.43)--(1.64,0.52));  draw((2.41,2.48)--(-2.48,0.52));  dot((-0.16,3.1),dotstyle);  label("$A$", (-0.4,3.3), NE * labelscalefactor);  dot((-2.48,0.52),dotstyle);  label("$B$", (-2.84,0.54), SW * labelscalefactor);  dot((3.78,0.52),dotstyle);  label("$C$", (3.86,0.64), NE * labelscalefactor);  dot((1.4,2.08),dotstyle);  label("$D$", (1.48,2.2), NE * labelscalefactor);  dot((1.64,0.52),dotstyle);  label("$E$", (1.72,0.64), NE * labelscalefactor);  dot((1.04,4.43),dotstyle);  label("$F$", (1.12,4.56), NE * labelscalefactor);  dot((2.41,2.48),dotstyle);  label("$M$", (2.48,2.6), NE * labelscalefactor);  clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);  [/asy]


See also

2003 USAMO (ProblemsResources)
Preceded by
Problem 3
Followed by
Problem 5
1 2 3 4 5 6
All USAMO Problems and Solutions

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