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Difference between revisions of "2019 USAJMO Problems/Problem 4"
(→‎Solution)
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{{MAA Notice}}
 
{{MAA Notice}}
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==Solution 2 ==
 +
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The answer is no.
 +
[asy]
 +
pair A, B, C, E, F, Ep, Fp, I, J, K, L;
 +
//A = dir(120);
 +
//B = 2*foot(origin, A, A+dir(250)) - A;
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//C = 2*foot(origin, A, A+dir(290)) - A;
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A = dir(155); B = dir(230); C = dir(310);
 +
E = foot(B, A, C);
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F = foot(C, A, B);
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I = incenter(A, B, C);
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J = 2*circumcenter(B, I, C) - I;
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K = foot(J, A, B);
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L = foot(J, A, C);
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Ep = reflect(A, I) * E;
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Fp = reflect(A, I) * F;
 +
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draw(B--K^^C--L, rgb(1, 0.7, 0.2));
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//draw(unitcircle, dotted);
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draw(A--B^^A--C, rgb(0.7, 0.3, 0));
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draw(B--C, heavycyan);
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draw(E--F, heavycyan);
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draw(Ep--Fp, heavycyan + dashed);
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filldraw(circle(J, abs(J-foot(J, B, C))), opacity(0.5) + pink, red);
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draw(B--E^^C--F, heavymagenta);
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draw(E--Ep^^F--Fp, dotted);
 +
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dot("<math>A</math>", A, dir(110));
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dot("<math>B</math>", B, dir(190));
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dot("<math>C</math>", C, dir(50));
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dot("<math>E</math>", E, dir(50));
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dot("<math>F</math>", F, dir(190));
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dot("<math>E'</math>", Ep, dir(190));
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dot("<math>F'</math>", Fp, dir(50));
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[/asy]
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 +
Suppose otherwise. Consider the reflection over the bisector of <math>\angle BAC</math>. This swaps rays <math>AB</math> and <math>AC</math>; suppose <math>E</math> and <math>F</math> are sent to <math>E'</math> and <math>F'</math>. Note that the <math>A</math>-excircle is fixed, so line <math>E'F'</math> must also be tangent to the <math>A</math>-excircle.
 +
 +
Since <math>BEFC</math> is cyclic, we obtain <math>\measuredangle ECB = \measuredangle EFB = \measuredangle EF'E'</math>, so <math>\overline{E'F'} \parallel \overline{BC}</math>. However, as <math>\overline{EF}</math> is a chord in the circle with diameter <math>\overline{BC}</math>, <math>EF \le BC</math>.
 +
 +
If <math>EF < BC</math> then <math>E'F' < BC</math> too, so then <math>\overline{E'F'}</math> lies inside <math>\triangle ABC</math> and cannot be tangent to the excircle.
 +
 +
The remaining case is when <math>EF = BC</math>. In this case, <math>\overline{EF}</math> is also a diameter, so <math>BECF</math> is a rectangle. In particular <math>\overline{BE} \parallel \overline{CF}</math>. However, by the existence of the orthocenter, the lines <math>BE</math> and <math>CF</math> must intersect, contradiction.
  
 
==See also==
 
==See also==
 
{{USAJMO newbox|year=2019|num-b=3|num-a=5}}
 
{{USAJMO newbox|year=2019|num-b=3|num-a=5}}

Revision as of 18:08, 19 April 2019

$(*)$ Let $ABC$ be a triangle with $\angle ABC$ obtuse. The [i]$A$-excircle[/i] is a circle in the exterior of $\triangle ABC$ that is tangent to side $\overline{BC}$ of the triangle and tangent to the extensions of the other two sides. Let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to lines $AC$ and $AB$, respectively. Can line $EF$ be tangent to the $A$-excircle?

Solution

Instead of trying to find a synthetic way to describe $EF$ being tangent to the $A$-excircle (very hard), we instead consider the foot of the perpendicular from the $A$-excircle to $EF$, hoping to force something via the length of the perpendicular. It would be nice if there were an easier way to describe $EF$, something more closely related to the $A$-excircle; as we are considering perpendicularity, if we could generate a line parallel to $EF$, that would be good.

So we recall that it is well known that triangle $AEF$ is similar to $ABC$. This motivates reflecting $BC$ over the angle bisector at $A$ to obtain $B'C'$, which is parallel to $EF$ for obvious reasons.

Furthermore, as reflection preserves intersection, $B'C'$ is tangent to the reflection of the $A$-excircle over the $A$-angle bisector. But it is well-known that the $A$-excenter lies on the $A$-angle bisector, so the $A$-excircle must be preserved under reflection over the $A$-excircle. Thus $B'C'$ is tangent to the $A$-excircle.Yet for all lines parallel to $EF$, there are only two lines tangent to the $A$-excircle, and only one possibility for $EF$, so $EF = B'C'$.

Thus as $ABB'$ is isoceles, \[[ABC] = \frac{1}{2} \cdot AC \cdot BE = \frac{AC}{2} \cdot \sqrt{AB^2 - AE^2} = \frac{AC}{2} \cdot \sqrt{AB^2 - AB'^2} = \frac{AC}{2} \cdot \sqrt{AB^2 - AB^2} = 0,\] contradiction. -alifenix-

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Solution 2

The answer is no. [asy] pair A, B, C, E, F, Ep, Fp, I, J, K, L; //A = dir(120); //B = 2*foot(origin, A, A+dir(250)) - A; //C = 2*foot(origin, A, A+dir(290)) - A; A = dir(155); B = dir(230); C = dir(310); E = foot(B, A, C); F = foot(C, A, B); I = incenter(A, B, C); J = 2*circumcenter(B, I, C) - I; K = foot(J, A, B); L = foot(J, A, C); Ep = reflect(A, I) * E; Fp = reflect(A, I) * F;

draw(B--K^^C--L, rgb(1, 0.7, 0.2)); //draw(unitcircle, dotted); draw(A--B^^A--C, rgb(0.7, 0.3, 0)); draw(B--C, heavycyan); draw(E--F, heavycyan); draw(Ep--Fp, heavycyan + dashed); filldraw(circle(J, abs(J-foot(J, B, C))), opacity(0.5) + pink, red); draw(B--E^^C--F, heavymagenta); draw(E--Ep^^F--Fp, dotted);

dot("$A$", A, dir(110)); dot("$B$", B, dir(190)); dot("$C$", C, dir(50)); dot("$E$", E, dir(50)); dot("$F$", F, dir(190)); dot("$E'$", Ep, dir(190)); dot("$F'$", Fp, dir(50)); [/asy]

Suppose otherwise. Consider the reflection over the bisector of $\angle BAC$. This swaps rays $AB$ and $AC$; suppose $E$ and $F$ are sent to $E'$ and $F'$. Note that the $A$-excircle is fixed, so line $E'F'$ must also be tangent to the $A$-excircle.

Since $BEFC$ is cyclic, we obtain $\measuredangle ECB = \measuredangle EFB = \measuredangle EF'E'$, so $\overline{E'F'} \parallel \overline{BC}$. However, as $\overline{EF}$ is a chord in the circle with diameter $\overline{BC}$, $EF \le BC$.

If $EF < BC$ then $E'F' < BC$ too, so then $\overline{E'F'}$ lies inside $\triangle ABC$ and cannot be tangent to the excircle.

The remaining case is when $EF = BC$. In this case, $\overline{EF}$ is also a diameter, so $BECF$ is a rectangle. In particular $\overline{BE} \parallel \overline{CF}$. However, by the existence of the orthocenter, the lines $BE$ and $CF$ must intersect, contradiction.

See also

2019 USAJMO (Problems • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions
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