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2017 USAJMO Problems/Problem 5: Difference between revisions

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Let <math>O</math> and <math>H</math> be the circumcenter and the orthocenter of an acute triangle <math>ABC</math>. Points <math>M</math> and <math>D</math> lie on side <math>BC</math> such that <math>BM = CM</math> and <math>\angle BAD = \angle CAD</math>. Ray <math>MO</math> intersects the circumcircle of triangle <math>BHC</math> in point <math>N</math>. Prove that <math>\angle ADO = \angle HAN</math>.  
Let <math>O</math> and <math>H</math> be the circumcenter and the orthocenter of an acute triangle <math>ABC</math>. Points <math>M</math> and <math>D</math> lie on side <math>BC</math> such that <math>BM = CM</math> and <math>\angle BAD = \angle CAD</math>. Ray <math>MO</math> intersects the circumcircle of triangle <math>BHC</math> in point <math>N</math>. Prove that <math>\angle ADO = \angle HAN</math>.  


==Solution==
==Solution 1==
[asy] pair A = dir(130); pair B = dir(220); pair C = dir(320); draw(unitcircle, lightblue);
<asy>
import olympiad;


pair P = dir(-90); pair Q = dir(90); pair D = extension(A, P, B, C); pair O = origin; pair M = extension(B, C, O, P); pair N = 2*M-P;
unitsize(100);


draw(A--B--C--cycle, lightblue); draw(A--P--Q, lightblue); draw(A--N--D--O--A, lightblue);
pair pA = dir(120);
pair pB = dir(225);
pair pC = dir(315);
pair pO = origin;
pair pH = orthocenter(pA, pB, pC);
pair pM = midpoint(pB--pC);
pair dD = bisectorpoint(pB, pA, pC);
pair pD = extension(pA, dD, pB, pC);
pair pN = intersectionpoints(pM--(3*pO-2*pM), circumcircle(pB, pH, pC))[0];
pair dHprime = foot(pH, pB, pC);
pair pHprime = 2*dHprime-pH;
pair dAprime = foot(pA, pB, pC);
pair pAprime = 2*dAprime-pA;
pair dNprime = foot(pN, pB, pC);
pair pNprime = 2*dNprime-pN;


draw(A--D--N--O--cycle, red);
draw(pA--pB--pC--cycle);
draw(unitcircle, blue);
draw(pH--pHprime, magenta);
draw(pA--pD, red);
draw(circumcircle(pB, pH, pC), blue);
draw(pA--pH, blue);
draw(pA--pN, magenta);
draw(pA--pO, blue);
draw(pD--pNprime, magenta);
 
dot("\(A\)", pA, NW);
dot("\(B\)", pB, W);
dot("\(C\)", pC, E);
dot("\(O\)", pO, SW, blue);
dot("\(H\)", pH, SW, blue);
dot("\(N\)", pN, NE, magenta);
dot("\(M\)", pM, S, red);
dot("\(D\)", pD, S, red);
dot("\(H'\)", pHprime, SW, magenta);
dot("\(A'\)", pAprime, SW);
dot("\(N'\)", pNprime, S, magenta);
</asy>
(original diagram by [[User:integralarefun|integralarefun]])
 
It's well known that the reflection of <math>H</math> across <math>\overline{BC}</math>, <math>H'</math>, lies on <math>(ABC)</math>. Then <math>(BHC)</math> is just the reflection of <math>(BH'C)</math> across <math>\overline{BC}</math>, which is equivalent to the reflection of <math>(ABC)</math> across <math>\overline{BC}</math>. Reflect points <math>A</math> and <math>N</math> across <math>\overline{BC}</math> to points <math>A'</math> and <math>N'</math>, respectively. Then <math>N'</math> is the midpoint of minor arc <math>\overarc{BC}</math>, so <math>A, D, N'</math> are collinear in that order. It suffices to show that <math>\angle AA'N'=\angle ADO</math>.


dot("<math>A</math>", A, dir(A)); dot("<math>B</math>", B, dir(B)); dot("<math>C</math>", C, dir(C)); dot("<math>P</math>", P, dir(P)); dot("<math>Q</math>", Q, dir(Q)); dot("<math>D</math>", D, dir(225)); dot("<math>O</math>", O, dir(315)); dot("<math>M</math>", M, dir(315)); dot("<math>N</math>", N, dir(315));
<b>Claim:</b> <math>\triangle AA'N' \sim \triangle ADO</math>. The proof easily follows.


/* TSQ Source:
<b>Proof:</b> Note that <math>\angle BAA'=\angle CAO=90^{\circ}-\angle ABC</math>. Then we have <math>\angle A'AN'=\angle BAD-\angle BAA'=\angle CAD-\angle CAO=\angle DAO</math>. So, it suffices to show that <cmath>\frac{AA'}{AN'}=\frac{AD}{AO}\rightarrow AA'\cdot AO=AN'\cdot AD.</cmath> Notice that <math>\triangle ABA' \sim \triangle AOC</math>, so that <cmath>\frac{AB}{AA'}=\frac{AO}{AC}\rightarrow AA'\cdot AO=AB\cdot AC.</cmath> Therefore, it suffices to show that <cmath>AB\cdot AC=AN'\cdot AD\rightarrow \frac{AB}{AN'}=\frac{AD}{AC}.</cmath> But it is easy to show that <math>\triangle BAN'\sim \triangle DAC</math>, implying the result. <math>\blacksquare</math>


A = dir 130 B = dir 220 C = dir 320 unitcircle 0.1 lightcyan / lightblue
==Solution 2==
<asy>
size(9cm);
pair A = dir(130);
pair B = dir(220);
pair C = dir(320);
draw(unitcircle, lightblue);


P = dir -90 Q = dir 90 D = extension A P B C R225 O = origin R315 M = extension B C O P R315 N = 2*M-P R315
pair P = dir(-90);
pair Q = dir(90);
pair D = extension(A, P, B, C);
pair O = origin;
pair M = extension(B, C, O, P);
pair N = 2*M-P;


A--B--C--cycle lightblue A--P--Q lightblue A--N--D--O--A lightblue
draw(A--B--C--cycle, lightblue);
draw(A--P--Q, lightblue);
draw(A--N--D--O--A, lightblue);


A--D--N--O--cycle 0.1 yellow / red
draw(A--D--N--O--cycle, red);


[/asy]
dot("$A$", A, dir(A));
dot("$B$", B, dir(B));
dot("$C$", C, dir(C));
dot("$P$", P, dir(P));
dot("$Q$", Q, dir(Q));
dot("$D$", D, dir(225));
dot("$O$", O, dir(315));
dot("$M$", M, dir(315));
dot("$N$", N, dir(315));
</asy>


Suppose ray <math>OM</math> intersects the circumcircle of <math>BHC</math> at <math>N'</math>, and let the foot of the A-altitude of <math>ABC</math> be <math>E</math>. Note that <math>\angle BHE=90-\angle HBE=90-90+\angle C=\angle C</math>. Likewise, <math>\angle CHE=\angle B</math>. So, <math>\angle BHC=\angle BHE+\angle CHE=\angle B+\angle C</math>.
Suppose ray <math>OM</math> intersects the circumcircle of <math>BHC</math> at <math>N'</math>, and let the foot of the A-altitude of <math>ABC</math> be <math>E</math>. Note that <math>\angle BHE=90-\angle HBE=90-90+\angle C=\angle C</math>. Likewise, <math>\angle CHE=\angle B</math>. So, <math>\angle BHC=\angle BHE+\angle CHE=\angle B+\angle C</math>.

Latest revision as of 11:05, 7 June 2024

Problem

Let $O$ and $H$ be the circumcenter and the orthocenter of an acute triangle $ABC$. Points $M$ and $D$ lie on side $BC$ such that $BM = CM$ and $\angle BAD = \angle CAD$. Ray $MO$ intersects the circumcircle of triangle $BHC$ in point $N$. Prove that $\angle ADO = \angle HAN$.

Solution 1

[asy] import olympiad; unitsize(100); pair pA = dir(120); pair pB = dir(225); pair pC = dir(315); pair pO = origin; pair pH = orthocenter(pA, pB, pC); pair pM = midpoint(pB--pC); pair dD = bisectorpoint(pB, pA, pC); pair pD = extension(pA, dD, pB, pC); pair pN = intersectionpoints(pM--(3*pO-2*pM), circumcircle(pB, pH, pC))[0]; pair dHprime = foot(pH, pB, pC); pair pHprime = 2*dHprime-pH; pair dAprime = foot(pA, pB, pC); pair pAprime = 2*dAprime-pA; pair dNprime = foot(pN, pB, pC); pair pNprime = 2*dNprime-pN; draw(pA--pB--pC--cycle); draw(unitcircle, blue); draw(pH--pHprime, magenta); draw(pA--pD, red); draw(circumcircle(pB, pH, pC), blue); draw(pA--pH, blue); draw(pA--pN, magenta); draw(pA--pO, blue); draw(pD--pNprime, magenta); dot("\(A\)", pA, NW); dot("\(B\)", pB, W); dot("\(C\)", pC, E); dot("\(O\)", pO, SW, blue); dot("\(H\)", pH, SW, blue); dot("\(N\)", pN, NE, magenta); dot("\(M\)", pM, S, red); dot("\(D\)", pD, S, red); dot("\(H'\)", pHprime, SW, magenta); dot("\(A'\)", pAprime, SW); dot("\(N'\)", pNprime, S, magenta); [/asy] (original diagram by integralarefun)

It's well known that the reflection of $H$ across $\overline{BC}$, $H'$, lies on $(ABC)$. Then $(BHC)$ is just the reflection of $(BH'C)$ across $\overline{BC}$, which is equivalent to the reflection of $(ABC)$ across $\overline{BC}$. Reflect points $A$ and $N$ across $\overline{BC}$ to points $A'$ and $N'$, respectively. Then $N'$ is the midpoint of minor arc $\overarc{BC}$, so $A, D, N'$ are collinear in that order. It suffices to show that $\angle AA'N'=\angle ADO$.

Claim: $\triangle AA'N' \sim \triangle ADO$. The proof easily follows.

Proof: Note that $\angle BAA'=\angle CAO=90^{\circ}-\angle ABC$. Then we have $\angle A'AN'=\angle BAD-\angle BAA'=\angle CAD-\angle CAO=\angle DAO$. So, it suffices to show that \[\frac{AA'}{AN'}=\frac{AD}{AO}\rightarrow AA'\cdot AO=AN'\cdot AD.\] Notice that $\triangle ABA' \sim \triangle AOC$, so that \[\frac{AB}{AA'}=\frac{AO}{AC}\rightarrow AA'\cdot AO=AB\cdot AC.\] Therefore, it suffices to show that \[AB\cdot AC=AN'\cdot AD\rightarrow \frac{AB}{AN'}=\frac{AD}{AC}.\] But it is easy to show that $\triangle BAN'\sim \triangle DAC$, implying the result. $\blacksquare$

Solution 2

[asy] size(9cm); pair A = dir(130); pair B = dir(220); pair C = dir(320); draw(unitcircle, lightblue); pair P = dir(-90); pair Q = dir(90); pair D = extension(A, P, B, C); pair O = origin; pair M = extension(B, C, O, P); pair N = 2*M-P; draw(A--B--C--cycle, lightblue); draw(A--P--Q, lightblue); draw(A--N--D--O--A, lightblue); draw(A--D--N--O--cycle, red); dot("$A$", A, dir(A)); dot("$B$", B, dir(B)); dot("$C$", C, dir(C)); dot("$P$", P, dir(P)); dot("$Q$", Q, dir(Q)); dot("$D$", D, dir(225)); dot("$O$", O, dir(315)); dot("$M$", M, dir(315)); dot("$N$", N, dir(315)); [/asy]

Suppose ray $OM$ intersects the circumcircle of $BHC$ at $N'$, and let the foot of the A-altitude of $ABC$ be $E$. Note that $\angle BHE=90-\angle HBE=90-90+\angle C=\angle C$. Likewise, $\angle CHE=\angle B$. So, $\angle BHC=\angle BHE+\angle CHE=\angle B+\angle C$. $BHCN'$ is cyclic, so $\angle BN'C=180-\angle BHC=180-\angle B-\angle C=\angle A$. Also, $\angle BAC=\angle A$. These two angles are on different circles and have the same measure, but they point to the same line $BC$! Hence, the two circles must be congruent. (This is also a well-known result)

We know, since $M$ is the midpoint of $BC$, that $OM$ is perpendicular to $BC$. $AH$ is also perpendicular to $BC$, so the two lines are parallel. $AN$ is a transversal, so $\angle HAN=\angle ANO$. We wish to prove that $\angle ANO=\angle ADO$, which is equivalent to $AOND$ being cyclic.

Now, assume that ray $OM$ intersects the circumcircle of $ABC$ at a point $P$. Point $P$ must be the midpoint of $\stackrel{\frown}{BC}$. Also, since $AD$ is an angle bisector, it must also hit the circle at the point $P$. The two circles are congruent, which implies $MN=MP\implies ND=DP\implies$ NDP is isosceles. Angle ADN is an exterior angle, so $\angle ADN=\angle DNP+\angle DPO=2\angle DPO$. Assume WLOG that $\angle B>\angle C$. So, $\angle DPO=\angle APO=\frac{\angle B+\angle C}{2}-\angle C=\frac{\angle B-\angle C}{2}$. In addition, $\angle AON=\angle AOP=\angle AOB+\angle BOP=2\angle C+\angle A$. Combining these two equations, $\angle AON+\angle ADN=\angle B-\angle C+2\angle C+\angle A=\angle A+\angle B+\angle C=180$.

Opposite angles sum to $180$, so quadrilateral $AOND$ is cyclic, and the condition is proved.

-william122

See also

2017 USAJMO (Problems • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions
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