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Difference between revisions of "2019 USAJMO Problems/Problem 4"

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<math>(*)</math> Let <math>ABC</math> be a triangle with <math>\angle ABC</math> obtuse. The <math>A</math>''-excircle'' is a circle in the exterior of <math>\triangle ABC</math> that is tangent to side <math>\overline{BC}</math> of the triangle and tangent to the extensions of the other two sides. Let <math>E</math>, <math>F</math> be the feet of the altitudes from <math>B</math> and <math>C</math> to lines <math>AC</math> and <math>AB</math>, respectively. Can line <math>EF</math> be tangent to the <math>A</math>-excircle?
 
<math>(*)</math> Let <math>ABC</math> be a triangle with <math>\angle ABC</math> obtuse. The <math>A</math>''-excircle'' is a circle in the exterior of <math>\triangle ABC</math> that is tangent to side <math>\overline{BC}</math> of the triangle and tangent to the extensions of the other two sides. Let <math>E</math>, <math>F</math> be the feet of the altitudes from <math>B</math> and <math>C</math> to lines <math>AC</math> and <math>AB</math>, respectively. Can line <math>EF</math> be tangent to the <math>A</math>-excircle?
  
−
==Solution==
+
==Solution 1==
  
 
Instead of trying to find a synthetic way to describe <math>EF</math> being tangent to the <math>A</math>-excircle (very hard), we instead consider the foot of the perpendicular from the <math>A</math>-excircle to <math>EF</math>, hoping to force something via the length of the perpendicular.  It would be nice if there were an easier way to describe <math>EF</math>, something more closely related to the <math>A</math>-excircle; as we are considering perpendicularity, if we could generate a line parallel to <math>EF</math>, that would be good.   
 
Instead of trying to find a synthetic way to describe <math>EF</math> being tangent to the <math>A</math>-excircle (very hard), we instead consider the foot of the perpendicular from the <math>A</math>-excircle to <math>EF</math>, hoping to force something via the length of the perpendicular.  It would be nice if there were an easier way to describe <math>EF</math>, something more closely related to the <math>A</math>-excircle; as we are considering perpendicularity, if we could generate a line parallel to <math>EF</math>, that would be good.   
Line 23: Line 23:
  
 
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.
 
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.
 +
 +
==Solution 3==
 +
 +
The answer is <math>\boxed{\text{no}}</math>.
 +
 +
Suppose for the sake of contradiction that it is possible for <math>EF</math> to be tangent to the <math>A</math>-excircle. Call the tangency point <math>T</math>, and let <math>S_1, S_2</math> denote the contact points of <math>AB, AC</math> with the <math>A</math>-excircle, respectively. Let <math>s</math> denote the semiperimeter of <math>ABC</math>. By equal tangents, we have <cmath>ET = ES_2, FT = FS_1 \implies EF = ES_2+FS_2</cmath>It is also well known that <math>AS_1 = AS_2 = \frac{s}{2}</math>, so <cmath>EF = ES_2+FS_2 = (AS_2-AE)+(AS_1-AF) = s-AE-AF \implies s=AE+AF+EF</cmath>It is well known (by an easy angle chase) that <math>\triangle AEF \sim \triangle ABC</math>, so we must have the ratio of similitude is <math>2</math>. In particular, <cmath>AB=2 \cdot AE, AC=2 \cdot AF</cmath>This results in <cmath>\angle ABE = 30^{\circ}, \angle CBF = 30^{\circ} \implies \angle EBC = 120^{\circ}</cmath>which is absurd since <math>\triangle BEC</math> is a right triangle. We reached a contradiction, so we are done. <math>\blacksquare</math> ~ Mathscienceclass
 +
 +
==Solution 4==
 +
<asy>
 +
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
 +
import graph; size(10cm);
 +
real labelscalefactor = 0.5; /* changes label-to-point distance */
 +
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
 +
pen dotstyle = black; /* point style */
 +
real xmin = 0.92, xmax = 28.12, ymin = -14.66, ymax = 5.16;  /* image dimensions */
 +
pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen wvvxds = rgb(0.396078431372549,0.3411764705882353,0.8235294117647058); pen sexdts = rgb(0.1803921568627451,0.49019607843137253,0.19607843137254902);
 +
/* draw figures */
 +
draw((3.52,1.38)--(5.8,-4.7), linewidth(1) + rvwvcq);
 +
draw((5.8,-4.7)--(10.3,-4.9), linewidth(1) + wvvxds);
 +
draw((10.3,-4.9)--(3.52,1.38), linewidth(1) + rvwvcq);
 +
draw(circle((9.325581455949688,-7.26800412402583), 2.399418339060914), linewidth(1) + sexdts);
 +
draw((5.8,-4.7)--(7.0789360558927,-8.1104961490472), linewidth(1) + rvwvcq);
 +
draw((10.3,-4.9)--(10.956076160142976,-5.507692962492314), linewidth(1) + rvwvcq);
 +
draw((8.503617697888226,-3.2360942688404215)--(6.711506849315068,-7.130684931506849), linewidth(1) + wvvxds);
 +
/* dots and labels */
 +
dot((3.52,1.38),dotstyle);
 +
label("$A$", (3.6,1.58), NE * labelscalefactor);
 +
dot((5.8,-4.7),dotstyle);
 +
label("$B$", (5.88,-4.5), NE * labelscalefactor);
 +
dot((10.3,-4.9),dotstyle);
 +
label("$C$", (10.38,-4.7), NE * labelscalefactor);
 +
dot((9.325581455949688,-7.26800412402583),dotstyle);
 +
label("$I_a$", (9.4,-7.06), NE * labelscalefactor);
 +
dot((7.0789360558927,-8.1104961490472),linewidth(4pt) + dotstyle);
 +
label("$G$", (7.16,-7.96), NE * labelscalefactor);
 +
dot((10.956076160142976,-5.507692962492314),linewidth(4pt) + dotstyle);
 +
label("$H$", (11.04,-5.34), NE * labelscalefactor);
 +
dot((8.503617697888226,-3.2360942688404215),dotstyle);
 +
label("E", (8.58,-3.04), NE * labelscalefactor);
 +
dot((6.711506849315068,-7.130684931506849),dotstyle);
 +
label("$F$", (6.8,-6.94), NE * labelscalefactor);
 +
dot((7.139200885553699,-6.201226130819783),dotstyle);
 +
label("$X$", (7.22,-6), NE * labelscalefactor);
 +
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 +
/* end of picture */
 +
</asy>
 +
We claim that the answer is no. We proceed with contradiction. Suppose that <math>EF</math> is indeed tangent to the a-excenter. Define the point of tangency to be <math>X</math>. Let <math>G</math> to be the intersection of the a-excircle with the extension of <math>AB</math> and <math>H</math> to be the intersection of the a-excircle with the extension of <math>AC</math>. Define <math>I_a</math> to be the a-excenter. It is a well-known fact that <math>I_a</math> lies on the angle bisector of <math>\angle BAC</math>.
 +
For convenience, let <math>AB = c, AC = b, BC = a</math>, <math>\angle CAB = A, \angle CBA = B, \angle ACB = C</math>.
 +
Notice that by Power of a Point:
 +
<cmath>FX = FG.</cmath>
 +
<cmath>XE = EH.</cmath>
 +
Therefore, adding, we see that:
 +
<cmath>FE = FX + XE = FG + EH.</cmath>
 +
It would be rather nice if we could re-write <math>FG</math>. Indeed, we can:
 +
<cmath>FG = AG - FG = \frac{r_a}{\tan \left( \frac{A}{2} \right)} - b \cos (A)</cmath>
 +
and similarly for <math>FH</math>:
 +
<cmath>FH = AH - AF = \frac{r_a}{\tan \left( \frac{A}{2} \right)} - c \cos (A)</cmath>
 +
We now seek to re-write <math>FE</math>. Using the law of cosines:
 +
<cmath>FE^2 = AF^2 + AE^ 2- 2AF \cdot AE \cos (A) = \left(c\cos (A) \right)^2 + \left(b\cos (A) \right)^2 -  2 (bc \cos ^2 (A)) \cos (A) = \cos ^2 (A) (c^2 + b^2 - 2bc \cos (A)) = \cos ^2 (A) \cdot a^2</cmath>
 +
Therefore,
 +
<cmath>FE = a \cos (A)</cmath>
 +
Putting this all together, we see that:
 +
<cmath>a \cos (A) = \frac{r_a}{\tan \left(\frac{a}{2} \right)} - b \cos (A) + \frac{r_a}{\tan \left(\frac{a}{2} \right)} - c \cos (A) \Longleftrightarrow</cmath>
 +
<cmath>(a+b+c) \cos (A) = \frac{2r_a}{\tan \left(\frac{a}{2} \right)}</cmath>
 +
Now, we seek to write <math>\tan \left(\frac{a}{2} \right)</math> in terms of the side lengths of the triangle. Notice that:
 +
<cmath>2\cos ^2 \left(\frac{A}{2} \right)-1 = \cos (A) \Longleftrightarrow</cmath>
 +
<cmath>2\cos ^2 \left(\frac{A}{2} \right) = 1 + \frac{b^2 + c^2-a^2}{2bc} = \frac{b^2 + 2bc + c^2 - a^2}{2bc} = \frac{(b+c)^2-a^2}{2bc} = \frac{(a+b+c)(b+c-a)}{2bc} = \frac{(2s)(2s-2a)}{2bc}</cmath>
 +
where <math>s</math> is the semi-perimeter. We get that:
 +
<cmath>\cos \left(\frac{A}{2} \right) = \sqrt{\frac{s(s-a)}{bc}}</cmath>
 +
Using the fact that <math>\cos^2 \left(\frac{A}{2} \right) + \sin^2 \left(\frac{A}{2} \right) = 1</math>, we have that:
 +
<cmath>\sin^2 \left(\frac{A}{2} \right) = 1-\frac{b^2 + 2bc + c^2 - a^2}{4bc} = \frac{(b-c)^2 - a^2}{4bc} = \frac{(2s-2c)(2s-2b)}{4bc}</cmath>
 +
Therefore,
 +
<cmath>\tan \left( \frac{A}{2} \right) = \sqrt{ \frac{(s-b)(s-c)}{s(s-a)} }</cmath>
 +
Returning to our original problem:
 +
<cmath>\frac{2r_a}{\sqrt{ \frac{(s-b)(s-c)}{s(s-a)} }} = (a+b+c) \cos (A)</cmath>
 +
It is a well-known fact that <math>r_a = \sqrt{\frac{s(s-b)(s-c)}{s-a}}</math>, so:
 +
<cmath>\frac{2\sqrt{\frac{s(s-b)(s-c)}{s-a}}}{\sqrt{ \frac{(s-b)(s-c)}{s(s-a)} }} = 2s \cos (A)</cmath> which implies that:
 +
<cmath>\cos (A) =1 \Longleftrightarrow A = 0</cmath>
 +
which is a contradiction. Hence, our original assumption that <math>EF</math> is tangent to the a-excircle is incorrect. <math>\blacksquare</math>
 +
~AopsUser101
  
 
==See also==
 
==See also==
 +
{{MAA Notice}}
 
{{USAJMO newbox|year=2019|num-b=3|num-a=5}}
 
{{USAJMO newbox|year=2019|num-b=3|num-a=5}}

Latest revision as of 19:27, 12 April 2021

Problem

$(*)$ Let $ABC$ be a triangle with $\angle ABC$ obtuse. The $A$-excircle 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 1

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-

Solution 2

The answer is no.

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.

Solution 3

The answer is $\boxed{\text{no}}$.

Suppose for the sake of contradiction that it is possible for $EF$ to be tangent to the $A$-excircle. Call the tangency point $T$, and let $S_1, S_2$ denote the contact points of $AB, AC$ with the $A$-excircle, respectively. Let $s$ denote the semiperimeter of $ABC$. By equal tangents, we have \[ET = ES_2, FT = FS_1 \implies EF = ES_2+FS_2\]It is also well known that $AS_1 = AS_2 = \frac{s}{2}$, so \[EF = ES_2+FS_2 = (AS_2-AE)+(AS_1-AF) = s-AE-AF \implies s=AE+AF+EF\]It is well known (by an easy angle chase) that $\triangle AEF \sim \triangle ABC$, so we must have the ratio of similitude is $2$. In particular, \[AB=2 \cdot AE, AC=2 \cdot AF\]This results in \[\angle ABE = 30^{\circ}, \angle CBF = 30^{\circ} \implies \angle EBC = 120^{\circ}\]which is absurd since $\triangle BEC$ is a right triangle. We reached a contradiction, so we are done. $\blacksquare$ ~ Mathscienceclass

Solution 4

[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = 0.92, xmax = 28.12, ymin = -14.66, ymax = 5.16; /* image dimensions */ pen rvwvcq = rgb(0.08235294117647059,0.396078431372549,0.7529411764705882); pen wvvxds = rgb(0.396078431372549,0.3411764705882353,0.8235294117647058); pen sexdts = rgb(0.1803921568627451,0.49019607843137253,0.19607843137254902); /* draw figures */ draw((3.52,1.38)--(5.8,-4.7), linewidth(1) + rvwvcq); draw((5.8,-4.7)--(10.3,-4.9), linewidth(1) + wvvxds); draw((10.3,-4.9)--(3.52,1.38), linewidth(1) + rvwvcq); draw(circle((9.325581455949688,-7.26800412402583), 2.399418339060914), linewidth(1) + sexdts); draw((5.8,-4.7)--(7.0789360558927,-8.1104961490472), linewidth(1) + rvwvcq); draw((10.3,-4.9)--(10.956076160142976,-5.507692962492314), linewidth(1) + rvwvcq); draw((8.503617697888226,-3.2360942688404215)--(6.711506849315068,-7.130684931506849), linewidth(1) + wvvxds); /* dots and labels */ dot((3.52,1.38),dotstyle); label("$A$", (3.6,1.58), NE * labelscalefactor); dot((5.8,-4.7),dotstyle); label("$B$", (5.88,-4.5), NE * labelscalefactor); dot((10.3,-4.9),dotstyle); label("$C$", (10.38,-4.7), NE * labelscalefactor); dot((9.325581455949688,-7.26800412402583),dotstyle); label("$I_a$", (9.4,-7.06), NE * labelscalefactor); dot((7.0789360558927,-8.1104961490472),linewidth(4pt) + dotstyle); label("$G$", (7.16,-7.96), NE * labelscalefactor); dot((10.956076160142976,-5.507692962492314),linewidth(4pt) + dotstyle); label("$H$", (11.04,-5.34), NE * labelscalefactor); dot((8.503617697888226,-3.2360942688404215),dotstyle); label("E", (8.58,-3.04), NE * labelscalefactor); dot((6.711506849315068,-7.130684931506849),dotstyle); label("$F$", (6.8,-6.94), NE * labelscalefactor); dot((7.139200885553699,-6.201226130819783),dotstyle); label("$X$", (7.22,-6), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy] We claim that the answer is no. We proceed with contradiction. Suppose that $EF$ is indeed tangent to the a-excenter. Define the point of tangency to be $X$. Let $G$ to be the intersection of the a-excircle with the extension of $AB$ and $H$ to be the intersection of the a-excircle with the extension of $AC$. Define $I_a$ to be the a-excenter. It is a well-known fact that $I_a$ lies on the angle bisector of $\angle BAC$. For convenience, let $AB = c, AC = b, BC = a$, $\angle CAB = A, \angle CBA = B, \angle ACB = C$. Notice that by Power of a Point: \[FX = FG.\] \[XE = EH.\] Therefore, adding, we see that: \[FE = FX + XE = FG + EH.\] It would be rather nice if we could re-write $FG$. Indeed, we can: \[FG = AG - FG = \frac{r_a}{\tan \left( \frac{A}{2} \right)} - b \cos (A)\] and similarly for $FH$: \[FH = AH - AF = \frac{r_a}{\tan \left( \frac{A}{2} \right)} - c \cos (A)\] We now seek to re-write $FE$. Using the law of cosines: \[FE^2 = AF^2 + AE^ 2- 2AF \cdot AE \cos (A) = \left(c\cos (A) \right)^2 + \left(b\cos (A) \right)^2 - 2 (bc \cos ^2 (A)) \cos (A) = \cos ^2 (A) (c^2 + b^2 - 2bc \cos (A)) = \cos ^2 (A) \cdot a^2\] Therefore, \[FE = a \cos (A)\] Putting this all together, we see that: \[a \cos (A) = \frac{r_a}{\tan \left(\frac{a}{2} \right)} - b \cos (A) + \frac{r_a}{\tan \left(\frac{a}{2} \right)} - c \cos (A) \Longleftrightarrow\] \[(a+b+c) \cos (A) = \frac{2r_a}{\tan \left(\frac{a}{2} \right)}\] Now, we seek to write $\tan \left(\frac{a}{2} \right)$ in terms of the side lengths of the triangle. Notice that: \[2\cos ^2 \left(\frac{A}{2} \right)-1 = \cos (A) \Longleftrightarrow\] \[2\cos ^2 \left(\frac{A}{2} \right) = 1 + \frac{b^2 + c^2-a^2}{2bc} = \frac{b^2 + 2bc + c^2 - a^2}{2bc} = \frac{(b+c)^2-a^2}{2bc} = \frac{(a+b+c)(b+c-a)}{2bc} = \frac{(2s)(2s-2a)}{2bc}\] where $s$ is the semi-perimeter. We get that: \[\cos \left(\frac{A}{2} \right) = \sqrt{\frac{s(s-a)}{bc}}\] Using the fact that $\cos^2 \left(\frac{A}{2} \right) + \sin^2 \left(\frac{A}{2} \right) = 1$, we have that: \[\sin^2 \left(\frac{A}{2} \right) = 1-\frac{b^2 + 2bc + c^2 - a^2}{4bc} = \frac{(b-c)^2 - a^2}{4bc} = \frac{(2s-2c)(2s-2b)}{4bc}\] Therefore, \[\tan \left( \frac{A}{2} \right) = \sqrt{ \frac{(s-b)(s-c)}{s(s-a)} }\] Returning to our original problem: \[\frac{2r_a}{\sqrt{ \frac{(s-b)(s-c)}{s(s-a)} }} = (a+b+c) \cos (A)\] It is a well-known fact that $r_a = \sqrt{\frac{s(s-b)(s-c)}{s-a}}$, so: \[\frac{2\sqrt{\frac{s(s-b)(s-c)}{s-a}}}{\sqrt{ \frac{(s-b)(s-c)}{s(s-a)} }} = 2s \cos (A)\] which implies that: \[\cos (A) =1 \Longleftrightarrow A = 0\] which is a contradiction. Hence, our original assumption that $EF$ is tangent to the a-excircle is incorrect. $\blacksquare$ ~AopsUser101

See also

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

2019 USAJMO (Problems • Resources)
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