Difference between revisions of "2016 USAJMO Problems/Problem 5"

Latest revision as of 18:15, 14 June 2020

Problem

Let $\triangle ABC$ be an acute triangle, with $O$ as its circumcenter. Point $H$ is the foot of the perpendicular from $A$ to line $\overleftrightarrow{BC}$ , and points $P$ and $Q$ are the feet of the perpendiculars from $H$ to the lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{AC}$ , respectively.

Given that $AH^2=2\cdot AO^2,$ prove that the points $O,P,$ and $Q$ are collinear.

Solution 1

$[asy] size(8cm); pair O=(0,0); pair A=dir(110); pair B=dir(-29); pair C=dir(209); pair H=foot(A,B,C); pair P=foot(H,A,B); pair Q=foot(H,A,C); draw(A--B--C--A--H--P); draw(circle(O,1)); draw(Q--H); dot("$A$", A, dir(A)); dot("$B$", B, dir(B)); dot("$C$", C, dir(C)); dot("$H$", H, S); dot("$P$", P, NE); dot("$Q$", Q, NW); dot("$O$", O, S); [/asy]$

It is well-known that $AH\cdot 2AO=AB\cdot AC$ (just use similar triangles or standard area formulas). Then by Power of a Point, $AP\cdot AB=AH^2=AQ\cdot AC$ Consider the transformation $X\mapsto \Psi(X)$ which dilates $X$ from $A$ by a factor of $\dfrac{AB}{AQ}=\dfrac{AC}{AP}$ and reflects about the $A$ -angle bisector. Then $\Psi(O)$ clearly lies on $AH$ , and its distance from $A$ is $AO\cdot\frac{AB}{AQ}=AO\cdot\frac{AB}{\frac{AH^2}{AC}}=AO\cdot\frac{AB\cdot AC}{AH^2}=\frac{AO\cdot AH\cdot 2AO}{AH^2}=\frac{2AO^2}{AH}=AH$ so $\Psi(O)=H$ , hence we conclude that $O,P,Q$ are collinear, as desired.

Solution 2

We will use barycentric coordinates with respect to $\triangle ABC.$ The given condition is equivalent to $(\sin B\sin C)^2=\frac{1}{2}.$ Note that $O=(\sin(2A):\sin(2B):\sin(2C)), P=(\cos^2B,\sin^2B,0), Q=(\cos^2C,0,\sin^2C).$ Therefore, we must show that $\begin{vmatrix} \sin(2A) & \sin(2B) & \sin(2C) \\ \cos^2B & \sin^2B & 0 \\ \cos^2C & 0 & \sin^2C \\ \end{vmatrix}=0.$ Expanding, we must prove $\sin(2A)\sin^2B\sin^2C=\cos^2C\sin^2B\sin(2C)+\sin^2C\cos^2B\sin(2B)$ $\frac{\sin(2A)}{2}=\sin^2B(1-\sin^2C)\sin(2C)+\sin^2C(1-\sin^2B)\sin(2B)$ $\begin{align*} \frac{\sin(2A)+\sin(2B)+\sin(2C)}{2}&=\sin^2B\sin(2C)+\sin^2C\sin(2B)\\ &=2\sin B\sin C(\sin B\cos C+\cos B\sin C) \\ &=2\sin B\sin C\sin A.\end{align*}$

Let $x=e^{iA}, y=e^{iB}, z=e^{iC},$ such that $xyz=-1.$ The left side is equal to $\frac{x^2+y^2+z^2-\frac{1}{x^2}-\frac{1}{y^2}-\frac{1}{z^2}}{4i}.$ The right side is equal to $\begin{align*} 2\cdot \frac{x-\frac{1}{x}}{2i}\cdot \frac{y-\frac{1}{y}}{2i}\cdot \frac{z-\frac{1}{z}}{2i}&=\frac{xyz-\frac{1}{xyz}-\frac{xy}{z}-\frac{yz}{x}-\frac{xz}{y}+\frac{x}{yz}+\frac{y}{xz}+\frac{z}{xy}}{-4i}\\ &=\frac{\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}-x^2-y^2-z^2}{-4i},\end{align*}$ which is equivalent to the left hand side. Therefore, the determinant is $0,$ and $O,P,Q$ are collinear. $\blacksquare$

Solution 3

For convenience, let $a, b, c$ denote the lengths of segments $BC, CA, AB,$ respectively, and let $\alpha, \beta, \gamma$ denote the measures of $\angle CAB, \angle ABC, \angle BCA,$ respectively. Let $R$ denote the circumradius of $\triangle ABC.$

Since the central angle $\angle AOB$ subtends the same arc as the inscribed angle $\angle ACB$ on the circumcircle of $\triangle ABC,$ we have $\angle AOB = 2\gamma.$ Note that $OA = OB,$ so $\angle OAB = \angle OBA.$ Thus, $\angle OAB = \frac{\pi}{2} - \gamma.$ Similarly, one can show that $\angle OAC = \frac{\pi}{2} - \beta.$ (One could probably cite this as well-known, but I have proved it here just in case.)

Clearly, $AO = R.$ Since $AH^2 = 2\cdot AO^2,$ we have $AH = \sqrt{2}R.$ Thus, $AH\cdot AO = \sqrt{2}R^2.$

Note that $AH = b\sin\gamma = c\sin\beta.$ The Extended Law of Sines states that: $\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma} = 2R.$ Therefore, $AH = \frac{bc}{2R} = \sqrt{2}R.$ Thus, $bc = \sqrt{2}R^2.$

Since $\angle PHA = \beta$ and $\angle QHA = \gamma,$ we have: $AP = AH\sin\beta = c\sin^2\beta = \frac{b^2 c}{4R^2}$ $AQ = AH\sin\gamma = b\sin^2\gamma = \frac{bc^2}{4R^2}$ It follows that: $AP\cdot AQ = \frac{b^3 c^3}{16R^4} = \frac{16\sqrt{2}R^6}{16R^4} = \sqrt{2}R^2.$ We see that $AP\cdot AQ = AH\cdot AO.$

Rearranging $AP\cdot AQ = AH\cdot AO,$ we get $\frac{AP}{AH} = \frac{AO}{AQ}.$ We also have $\angle PAH = \angle OAQ = \frac{\pi}{2} - \beta,$ so $\triangle PAH\sim\triangle OAQ$ by SAS similarity. Thus, $\angle AOQ = \angle APH,$ so $\angle AOQ$ is a right angle.

Rearranging $AP\cdot AQ = AH\cdot AO,$ we get $\frac{AP}{AO} = \frac{AO}{AH}.$ We also have $\angle PAO = \angle HAQ = \frac{\pi}{2} - \gamma,$ so $\triangle PAO\sim\triangle HAQ$ by SAS similarity. Thus, $\angle AOP = \angle AQH,$ so $\angle AOP$ is a right angle.

Since $\angle AOP$ and $\angle AOQ$ are both right angles, we get $\angle POQ = \pi,$ so we conclude that $P, O, Q$ are collinear, and we are done. (We also obtain the extra interesting fact that $AO\perp PQ.$ )

Solution 4

Draw the altitude from $O$ to $AB$ , and let the foot of this altitude be $D$ .

Then, by the Right Triangle Altitude Theorem on triangle $AHB$ , we have: $AB\cdot AP=AH^{2}$ .

Since $OD$ is the perpendicular bisector of $AB$ , $2\cdot AD = AB$ .

Substituting this into our previous equation gives $2\cdot AD \cdot AP = AH^{2}$ , which equals $2\cdot AO^{2}$ by the problem condition.

Thus, $2\cdot AD\cdot AP = 2\cdot AO^{2} \implies AD\cdot AP = AO^{2}$ .

Again, by the Right Triangle Altitude Theorem, angle $AOP$ is right.

By dropping an altitude from $O$ to $AC$ and using the same method, we can find that angle $AOQ$ is right. Since $\angle AOP=\angle AOQ=90$ , $P$ , $O$ , $Q$ are collinear and we are done.

~champion999

Solution 5

We use complex numbers. Let lower case letters represent their respective upper case points, with $|a| = |b| = |c| = 1$ . Spamming the foot from point to segment formula, we obtain $h = \dfrac{1}{2}\left(a+b+c-\dfrac{bc}{a}\right),$ $p = \dfrac{1}{2}(h+a+b-ab\overline{h}) = \dfrac{1}{4}\left(2a+2b+c-\dfrac{bc}{a}-\dfrac{ab}{c}+\dfrac{a^2}{c}\right),$ and $q = \dfrac{1}{2}(h+a+c-ac\overline{h}) = \dfrac{1}{4}\left(2a+b+2c-\dfrac{bc}{a}-\dfrac{ac}{b}+\dfrac{a^2}{b}\right).$ We now simplify the given length condition: $\begin{align*} AH^2 &= 2AO^2 \\ \implies (h-a)\overline{(h-a)} &= 2 \\ \implies \dfrac{1}{2}\left(-a+b+c-\dfrac{bc}{a}\right) \cdot \dfrac{1}{2}\left(-\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}-\dfrac{a}{bc}\right) &= 2 \\ \implies (ab+ac-bc-a^2)^2 &= 8a^2bc \\ \implies (a-b)^2(a-c)^2 &= 8a^2bc.\end{align*}$ We would like to show that $P$ , $O$ , $Q$ are collinear, or $\begin{vmatrix} p & \overline{p} & 1 \\ 0 & 0 & 1 \\ q & \overline{q} & 1 \end{vmatrix} = \overline{p}q - p\overline{q} = 0.$ After some factoring (or expanding) that takes about 15 minutes, this eventually reduces to $\overline{p}q - p\overline{q} = 0 \iff (a-b)(a-c)(b-c)((a-b)^2(a-c)^2 - 8a^2bc) = 0,$ which is true. $\square$

-MP8148

Solution 6

Claim: $\triangle AOQ \sim \triangle AHB$

Proof: We compute the area of $\triangle ABC$ using two methods. Let $R=AO$ be the circumradius of $\triangle ABC$ .

First, by extended law of sines, $BC=2R \sin \angle BAC$ . We are also given that $AH= R \sqrt{2}$ . $AH \perp BC$ , so $[ABC]=\frac{2R \sin \angle BAC \cdot R \sqrt{2}}{2}=R^2 \sin \angle BAC \sqrt{2}.$

Second, we compute the area using $[ABC]=\frac{1}{2} \cdot AB \cdot AC \sin \angle BAC$ .

Equating these two expressions for the area of $\triangle ABC$ and reducing, we get $AB \cdot AC = 2 R^2 \sqrt{2}.$ But $2R^2 =2 AO^2 = AH^2$ , so $AB \cdot AC = AH^2 \sqrt{2}$ and $\bf{AB =\frac{AH^2 \sqrt{2}}{AC}}$ .

Since $\angle AQH=\angle AHC =90^\circ$ , and both triangles share the angle $\angle HAC$ , $\triangle AHC \sim \triangle AQH$ . This tells us $\frac{AQ}{AH}=\frac{AH}{AC}$ or $AQ =\frac{AH^2}{AC}$ .

Substituting into the bolded equation, we get $AB=AQ \sqrt{2}$ .

The original length condition can be written as $AH= AO \sqrt{2}$ .

We also have $\angle OAQ= \angle OAC= 90- \angle ABC = \angle HAB.$

Therefore, by SAS similarity, $\triangle AOQ \sim \triangle AHB$ .

We can prove analogously that $\triangle AOP \sim \triangle AHC$ . We now have $\angle AOQ=\angle AHB =90^\circ$ and $\angle AOP=\angle AHC =90^\circ$ . This implies that $\angle POQ=\angle POA +\angle QOA=180^\circ$ which tells us $P,O,$ and $Q$ are collinear, as desired. $\square$

-vvluo

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

@@ Line 6: / Line 6: @@
 == Solution 1==
+<asy>
+size(8cm);
+pair O=(0,0);
+pair A=dir(110);
+pair B=dir(-29);
+pair C=dir(209);
+pair H=foot(A,B,C);
+pair P=foot(H,A,B);
+pair Q=foot(H,A,C);
+draw(A--B--C--A--H--P);
+draw(circle(O,1));
+draw(Q--H);
+dot("$A$", A, dir(A));
+dot("$B$", B, dir(B));
+dot("$C$", C, dir(C));
+dot("$H$", H, S);
+dot("$P$", P, NE);
+dot("$Q$", Q, NW);
+dot("$O$", O, S);
+</asy>
 It is well-known that <math>AH\cdot 2AO=AB\cdot AC</math> (just use similar triangles or standard area formulas). Then by Power of a Point,
 <cmath>AP\cdot AB=AH^2=AQ\cdot AC</cmath> Consider the transformation <math>X\mapsto \Psi(X)</math> which dilates <math>X</math> from <math>A</math> by a factor of <math>\dfrac{AB}{AQ}=\dfrac{AC}{AP}</math> and reflects about the <math>A</math>-angle bisector. Then <math>\Psi(O)</math> clearly lies on <math>AH</math>, and its distance from <math>A</math> is <cmath>AO\cdot\frac{AB}{AQ}=AO\cdot\frac{AB}{\frac{AH^2}{AC}}=AO\cdot\frac{AB\cdot AC}{AH^2}=\frac{AO\cdot AH\cdot 2AO}{AH^2}=\frac{2AO^2}{AH}=AH</cmath> so <math>\Psi(O)=H</math>, hence we conclude that <math>O,P,Q</math> are collinear, as desired.
@@ Line 67: / Line 89: @@
 ~champion999
+== Solution 5==
+We use complex numbers. Let lower case letters represent their respective upper case points, with <math>|a| = |b| = |c| = 1</math>. Spamming the foot from point to segment formula, we obtain <cmath>h = \dfrac{1}{2}\left(a+b+c-\dfrac{bc}{a}\right),</cmath> <cmath>p = \dfrac{1}{2}(h+a+b-ab\overline{h}) = \dfrac{1}{4}\left(2a+2b+c-\dfrac{bc}{a}-\dfrac{ab}{c}+\dfrac{a^2}{c}\right),</cmath> and <cmath>q = \dfrac{1}{2}(h+a+c-ac\overline{h}) = \dfrac{1}{4}\left(2a+b+2c-\dfrac{bc}{a}-\dfrac{ac}{b}+\dfrac{a^2}{b}\right).</cmath> We now simplify the given length condition: <cmath>\begin{align*} AH^2 &= 2AO^2 \\ \implies (h-a)\overline{(h-a)} &= 2 \\ \implies \dfrac{1}{2}\left(-a+b+c-\dfrac{bc}{a}\right) \cdot \dfrac{1}{2}\left(-\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}-\dfrac{a}{bc}\right) &= 2 \\ \implies (ab+ac-bc-a^2)^2 &= 8a^2bc \\ \implies (a-b)^2(a-c)^2 &= 8a^2bc.\end{align*}</cmath> We would like to show that <math>P</math>, <math>O</math>, <math>Q</math> are collinear, or <cmath>\begin{vmatrix} p & \overline{p} & 1 \\ 0 & 0 & 1 \\ q & \overline{q} & 1 \end{vmatrix} = \overline{p}q - p\overline{q} = 0.</cmath> After some factoring (or expanding) that takes about 15 minutes, this eventually reduces to <cmath>\overline{p}q - p\overline{q} = 0 \iff (a-b)(a-c)(b-c)((a-b)^2(a-c)^2 - 8a^2bc) = 0,</cmath> which is true. <math>\square</math>
+-MP8148
+==Solution 6==
+Claim: <math>\triangle AOQ \sim \triangle AHB</math>
+Proof: We compute the area of <math>\triangle ABC</math> using two methods. Let <math>R=AO</math> be the circumradius of <math>\triangle ABC</math>.
+First, by extended law of sines, <math>BC=2R \sin \angle BAC</math>. We are also given that <math>AH= R \sqrt{2}</math>.
+<math>AH \perp BC</math>, so <cmath>[ABC]=\frac{2R \sin \angle BAC \cdot R \sqrt{2}}{2}=R^2 \sin \angle BAC \sqrt{2}.</cmath>
+Second, we compute the area using <math>[ABC]=\frac{1}{2} \cdot AB \cdot AC \sin \angle BAC</math>.
+Equating these two expressions for the area of <math>\triangle ABC</math> and reducing, we get <cmath>AB \cdot AC = 2 R^2 \sqrt{2}.</cmath>
+But <math>2R^2 =2 AO^2 = AH^2</math>, so <math>AB \cdot AC = AH^2 \sqrt{2}</math> and <math>\bf{AB =\frac{AH^2 \sqrt{2}}{AC}}</math>.
+Since <math>\angle AQH=\angle AHC =90^\circ</math>, and both triangles share the angle <math>\angle HAC</math>, <math>\triangle AHC \sim \triangle AQH</math>. This tells us <math>\frac{AQ}{AH}=\frac{AH}{AC}</math> or <math>AQ =\frac{AH^2}{AC}</math>.
+Substituting into the bolded equation, we get <math>AB=AQ \sqrt{2}</math>.
+The original length condition can be written as <math>AH= AO \sqrt{2}</math>.
+We also have <cmath>\angle OAQ= \angle OAC= 90- \angle ABC = \angle HAB.</cmath>
+Therefore, by SAS similarity, <math>\triangle AOQ \sim \triangle AHB</math>.
+-----------------------------------------
+We can prove analogously that <math>\triangle AOP \sim \triangle AHC</math>.
+We now have <math>\angle AOQ=\angle AHB =90^\circ</math> and <math>\angle AOP=\angle AHC =90^\circ</math>.
+This implies that <math>\angle POQ=\angle POA +\angle QOA=180^\circ</math> which tells us <math>P,O,</math> and <math>Q</math> are collinear, as desired.<math>\square</math>
+-vvluo
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2016 USAJMO (Problems • Resources)
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