Difference between revisions of "2017 USAJMO Problems/Problem 3"

Revision as of 14:10, 10 May 2021

Problem

( $*$ ) Let $ABC$ be an equilateral triangle and let $P$ be a point on its circumcircle. Let lines $PA$ and $BC$ intersect at $D$ ; let lines $PB$ and $CA$ intersect at $E$ ; and let lines $PC$ and $AB$ intersect at $F$ . Prove that the area of triangle $DEF$ is twice that of triangle $ABC$ .

$[asy] size(3inch); pair A = (0, 3sqrt(3)), B = (-3,0), C = (3,0), P = (0, -sqrt(3)), D = (0, 0), E1 = (6, -3sqrt(3)), F = (-6, -3sqrt(3)), O = (0, sqrt(3)); draw(Circle(O, 2sqrt(3)), black); draw(A--B--C--cycle); draw(B--E1--C); draw(C--F--B); draw(A--P); draw(D--E1--F--cycle, dashed); label("A", A, N); label("B", B, W); label("C", C, E); label("P", P, S); label("D", D, NW); label("E", E1, SE); label("F", F, SW); [/asy]$

Solution (No Trig/Bash)

Extend $DP$ to hit $EF$ at $K$ . Then note that $[DEF]\cdot\frac{AK}{DK}\cdot\frac{AB}{AF}\cdot\frac{AC}{AE}=[ABC].$ Letting $BF=x$ and $PF=y$ , we have that $\frac{x+AB}y=\frac{y+PC}x=\frac{AC}{BP}.$ Solving and simplifying using LoC on $\triangle BPC$ gives $\frac{AB}{AF}=\frac{PC}{PB+PC}.$ Similarly, $\frac{AC}{AE}=\frac{PB}{PB+PC}.$

Now we find $\frac{AK}{DK}.$ Note that $\frac{AD}{DP}=\frac{AD}{BD}\cdot\frac{BD}{DP}=\frac{AC}{PB}\cdot\frac{AB}{PC}=\frac{AB^2}{PB\cdot PC}.$ Now let $E'=DE\cap AF$ and $F'=DF\cap AE$ . Then by an area/concurrence theorem, we have that $\frac{DK}{AK}+\frac{DE'}{EE'}+\frac{DF'}{FF'}=1,$ or $\frac{DK}{AK}+(1-\frac{DP}{AP}-\frac{DC}{BC})+(1-\frac{DP}{AP}-\frac{BD}{BC})=1.$ Thus we have that $\frac{DK}{AK}=2\cdot\frac{DP}{AP}.$

Manipulating these gives $\frac{AK}{DK}=\frac{(PB+PC)^2}{2\cdot PB\cdot PC}.$ Thus $\frac{AK}{DK}\cdot\frac{AB}{AF}\cdot\frac{AC}{AE}=\frac12,$ and we are done.

~cocohearts

Solution 1

WLOG, let $AB = 1$ . Let $[ABD] = X, [ACD] = Y$ , and $\angle BAD = \theta$ . After some angle chasing, we find that $\angle BCF \cong \angle BEC \cong \theta$ and $\angle FBC \cong \angle BCE \cong 120^{\circ}$ . Therefore, $\triangle FBC$ ~ $\triangle BCE$ .

Lemma 1: If $BF = k$ , then $CE = \frac 1k$ . This lemma results directly from the fact that $\triangle FBC$ ~ $\triangle BCE$ ; $\frac{BF}{BC} = \frac{BF}{1} = \frac{BC}{CE} = \frac{1}{CE}$ , or $CE = \frac{1}{BF}$ .

Lemma 2: $[AEF] = (k+\frac 1k + 2)(X+Y)$ . We see that $[AEF] = (X+Y) \frac{[AEF]}{[ABC]} = (k+1)(1+\frac 1k)(X+Y) = (k + \frac 1k + 2)(X+Y)$ , as desired.

Lemma 3: $\frac{X}{Y} = k$ . We see that $\frac XY = \frac{\frac 12 (AB)(AD) \sin(\theta)}{\frac 12 (AC)(AD) \sin (60^{\circ} - \theta)} = \frac{\sin(\theta)}{\sin (60^{\circ} - \theta)}.$ However, after some angle chasing and by the Law of Sines in $\triangle BCF$ , we have $\frac{k}{\sin(\theta)} = \frac{1}{\sin(60^{\circ} - \theta)}$ , or $k = \frac{\sin(\theta)}{\sin (60^{\circ} - \theta)}$ , which implies the result.

By the area lemma, we have $[BDF] = kX$ and $[CDF] = \frac{Y}{k}$ .

We see that $[DEF] = [AEF] - [ABC] - [BDF] - [DCE] = Xk + Yk + \frac Xk + \frac Yk + 2X + 2Y - X - Y - Xk - \frac Yk = X + Y + \frac Xk + yk$ . Thus, it suffices to show that $X + Y + \frac Xk + Yk = 2X + 2Y$ , or $\frac Xk + Yk = X + Y$ . Rearranging, we find this to be equivalent to $\frac XY = k$ , which is Lemma 3, so the result has been proven.

Solution 2

We will use barycentric coordinates and vectors. Let $\vec{X}$ be the position vector of a point $X.$ The point $(x, y, z)$ in barycentric coordinates denotes the point $x\vec{A} + y\vec{B} + z\vec{C}.$ For all points in the plane of $\triangle{ABC},$ we have $x + y + z = 1.$ It is clear that $A = (1, 0, 0)$ ; $B = (0, 1, 0)$ ; and $C = (0, 0, 1).$

Define the point $P$ as $P = \left(x_P, y_P, z_P\right).$ The fact that $P$ lies on the circumcircle of $\triangle{ABC}$ gives us $x^2_P + y^2_P + z^2_P = 1.$ This, along with the condition $x_P + y_P + z_P = 1$ inherent to barycentric coordinates, gives us $x_Py_P + y_Pz_P + z_Px_P = 0.$

We can write the equations of the following lines: $BC: x = 0$ $CA: y = 0$ $AB: z = 0$ $PA: \frac{y}{y_P} = \frac{z}{z_P}$ $PB: \frac{x}{x_P} = \frac{z}{z_P}$ $PC: \frac{x}{x_P} = \frac{y}{y_P}.$

We can then solve for the points $D, E, F$ : $D = \left(0, \frac{y_P}{y_P + z_P}, \frac{z_P}{y_P + z_P}\right)$ $E = \left(\frac{x_P}{x_P + z_P}, 0, \frac{z_P}{x_P + z_P}\right)$ $F = \left(\frac{x_P}{x_P + y_P}, \frac{y_P}{x_P + y_P}, 0\right).$

The area of an arbitrary triangle $XYZ$ is: $[XYZ] = \frac{1}{2}|\vec{XY}\times\vec{XZ}|$ $[XYZ] = \frac{1}{2}|(\vec{X}\times\vec{Y}) + (\vec{Y}\times\vec{Z}) + (\vec{Z}\times\vec{X})|.$

To calculate $[DEF],$ we wish to compute $(\vec{D}\times\vec{E}) + (\vec{E}\times\vec{F}) + (\vec{F}\times\vec{D}).$ After a lot of computation, we obtain the following: $(\vec{D}\times\vec{E}) + (\vec{E}\times\vec{F}) + (\vec{F}\times\vec{D}) = \frac{2x_Py_Pz_P}{(x_P + y_P)(y_P + z_P)(z_P + x_P)}[(\vec{A}\times\vec{B}) + (\vec{B}\times\vec{C}) + (\vec{C}\times\vec{A})].$

Evaluating the denominator, $(x_P + y_P)(y_P + z_P)(z_P + x_P) = (1 - z_P)(1 - y_P)(1 - x_P)$ $(x_P + y_P)(y_P + z_P)(z_P + x_P) = 1 - (x_P + y_P + z_P) + (x_Py_P + y_Pz_P + z_Px_P) - x_Py_Pz_P.$

Since $x_P + y_P + z_P = 1$ and $x_Py_P + y_Pz_P + z_Px_P = 0,$ it follows that: $(x_P + y_P)(y_P + z_P)(z_P + x_P) = -x_Py_Pz_P.$

We thus conclude that: $(\vec{D}\times\vec{E}) + (\vec{E}\times\vec{F}) + (\vec{F}\times\vec{D}) = -2[(\vec{A}\times\vec{B}) + (\vec{B}\times\vec{C}) + (\vec{C}\times\vec{A})].$

From this, it follows that $[DEF] = 2[ABC],$ and we are done.

Solution 3

$[asy] import cse5; import graph; import olympiad; size(3inch); pair A = (0, 3sqrt(3)), B = (-3,0), C = (3,0), O = (0, sqrt(3)); pair P = (-1, -sqrt(11)+sqrt(3)); path circle = Circle(O, 2sqrt(3)); pair D = extension(A,P,B,C); pair E1 = extension(A,C,B,P); pair F=extension(A,B,C,P); draw(circle, black); draw(A--B--C--cycle); draw(B--E1--C); draw(C--F--B); draw(A--P); draw(D--E1--F--cycle, dashed); pair G = extension(O,D,F,E1); draw(O--G,dashed); label("A", A, N); label("B", B, W); label("C", C, E); label("P", P, S); label("D", D, NW); label("E", E1, SE); label("F", F, SW); dot("O", O, SE); [/asy]$

We'll use coordinates and shoelace. Let the origin be the midpoint of $BC$ . Let $AB=2$ , and $BF = 2x$ , then $F=(-x-1,-x\sqrt{3})$ . Using the facts $\triangle{CBP} \sim \triangle{CFB}$ and $\triangle{BCP} \sim \triangle{BEC}$ , we have $BF * CE = BC^2$ , so $CE = \frac{1}{2x}$ , and $E = (\frac{1}{x}+1,-\frac{\sqrt{3}}{x})$ .

The slope of $FE$ is $k = \frac{-\frac{\sqrt{3}}{x} + x\sqrt{3}}{2+\frac{1}{x}+x}$ It is well-known that $\triangle{DFE}$ is self-polar, so $FE$ is the polar of $D$ , i.e., $OD$ is perpendicular to $FE$ . Therefore, the slope of $OD$ is $-\frac{1}{k}$ . Since $O=(0,\frac{1}{\sqrt{3}})$ , we get the x-coordinate of $D$ , $x_D = \frac{k}{\sqrt{3}}$ , i.e., $D = (\frac{k}{\sqrt{3}},0)$ . Using shoelace, $2[\triangle{FDE}] = \frac{k}{\sqrt{3}}(-x\sqrt{3})+(-x-1)(-\frac{\sqrt{3}}{x})- (-x\sqrt{3})(\frac{1}{x}+1) - (-\frac{\sqrt{3}}{x})\frac{k}{\sqrt{3}}$ $= 2\sqrt{3} + \sqrt{3}(\frac{1}{x}+x) + k(\frac{1}{x} - x)$ $= 2\sqrt{3} + \sqrt{3}(\frac{2(\frac{1}{x}+x)+(\frac{1}{x}+x)^2-(\frac{1}{x}-x)^2} {2+\frac{1}{x}+x})$ $= 2\sqrt{3} + \sqrt{3}(\frac{2(\frac{1}{x}+x)+4}{2+\frac{1}{x}+x})$ $= 4\sqrt{3}$ So $[\triangle{FDE}] = 2\sqrt{3} = 2[\triangle{ABC}]$ . Q.E.D

By Mathdummy.

Solution 4 Without the nasty computations

Note that $\angle{APB}=\angle{FPB}=\angle{EPC}=\angle{APC} = 60$ . We will use a special version of Stewart's theorem for angle bisectors in triangle with an 120 angle to calculate various side lengths.

Let $BP = x$ and $CP = y$ . Then, From Law of Cosine, $BC^2 = x^2 + y^2 + xy$ .

From Ptolemy's theorem, $AP BC = x AC + y AB$ , so $AP = x + y$ .

Lemma 1: In Triangle ABC with side lengths $a,b,c$ and $\angle A =120^o$ , the length of the angle bisector of $A$ is $d = \frac{bc}{b+c}$ This can be easily proved with Stewart's and Law of Cosine.

Using Lemma 1, we have $PD = \frac{xy}{x+y}$ $x = \frac{FP AP}{FP + AP}$ $y = \frac{EP AP}{EP + AP}$ Plug in $AP=x+y$ , we get: $PD = \frac{xy}{x+y}$ $FP = \frac{x(x+y)}{y}$ $EP = \frac{y(x+y)}{x}$ Then $[\triangle{FDE}] = \frac{1}{2}\sin(120)(PDFP + FPEP + EPPD)$ $= \frac{\sqrt{3}}{4}(x^2 + (x+y)^2 + y^2)$ $= \frac{\sqrt{3}}{2}(x^2 + xy + y^2)$ $= \frac{\sqrt{3}}{2} BC^2$ $= 2 [\triangle{ABC}]$

By Mathdummy.

Solution 5

Extend AP to meet FE at point X. Note that there is symmetry of triangle $\triangle{AFE}$ across line AX. Therefore, if we can prove that the area of $\triangle{FDX}$ is twice the area of $\triangle{BAD}$ , then we will be done. For simplicity, let the side length of equilateral triangle $\triangle{ABC}$ be $1$ . \\

We can easily find the area of $\triangle{BAD}$ . Since the area formula for an equilateral triangle is $\frac{s^2 \sqrt{3}}{4}$ , where $s$ is the side length, the area of $\triangle{ABC}$ is $\frac{\sqrt{3}}{4}$ . By symmetry, the area of $\triangle{BAD}$ is exactly half the area of $\triangle{ABC}$ . Thus the area of $\triangle{BAD} = \frac{\frac{\sqrt{3}}{4}}{2} = \boxed{\frac{\sqrt{3}}{8}}$ .

Now we just need to find the area of $\triangle{FDX}$ . We have to find $FX$ and $DX$ for the area. Note that $FX = \frac{FE}{2}$ . It looks by eye that $\triangle{AFE}$ is equilateral. Let's try to prove it.

\textbf{\textit{Lemma 1: $\triangle{AFE}$ is equilateral.}}

\textbf{\textit{Proof of Lemma 1:}} By symmetry, across lines $AX, FC, BE$ , we arrive to some conclusions: \begin{center}

       
       
       
       
       . 
       \end{center}
       
       Substituting the first equation into the second, we see that 					      , which is necessary and sufficient to prove that  is equilateral.  
       


Since  is equilateral, and so is , we find that . Because  by symmetry across , we know that the similarity ratio is .
Thus the side length of  is , and .



Now we need to find . We can do . Well, that's easy! Since  is a 30-60-90 triangle by symmetry across line , and , we know that . To find , we do pretty much the same thing:  is a 30-60-90 triangle, and , so . We can now find : 
\begin{center}
. 
\end{center}




Recalling why we were doing all this, we find the area of . It is .

Let's summarize: 

\begin{center}
The area of  is \\

The area of  is , \\

which proves our initial claim. 



\end{center}

@@ Line 182: / Line 182: @@
 We can easily find the area of <math>\triangle{BAD}</math>. Since the area formula for an equilateral triangle is <math>\frac{s^2 \sqrt{3}}{4}</math> , where <math>s</math> is the side length, the area of <math>\triangle{ABC}</math> is <math>\frac{\sqrt{3}}{4}</math>. By symmetry, the area of <math>\triangle{BAD}</math> is exactly half the area of <math>\triangle{ABC}</math>. Thus the area of <math>\triangle{BAD} = \frac{\frac{\sqrt{3}}{4}}{2} = \boxed{\frac{\sqrt{3}}{8}}</math>.
-Now we just need to find the area of <math>\triangle{FDX}</math>. We have to find <math>FX</math> and <math>DX</math> for the area. Note that <math>FX = \frac{FE}{2}</math>. It looks by eye that <math>\triangle{AFE}</math> is equilateral. Let's try to prove it. \\
+Now we just need to find the area of <math>\triangle{FDX}</math>. We have to find <math>FX</math> and <math>DX</math> for the area. Note that <math>FX = \frac{FE}{2}</math>. It looks by eye that <math>\triangle{AFE}</math> is equilateral. Let's try to prove it.
-\textbf{\textit{Lemma 1: <math>\triangle{AFE}</math> is equilateral.}}  \\
+\textbf{\textit{Lemma 1: <math>\triangle{AFE}</math> is equilateral.}}
 \textbf{\textit{Proof of Lemma 1:}}
@@ Line 200: / Line 200: @@
          Substituting the first equation into the second, we see that 					      <math>AF=AE=FE</math>, which is necessary and sufficient to prove that <math>\triangle{AFE}</math> is equilateral. <math>\square</math>
-         \\
   Since <math>\triangle{AFE}</math> is equilateral, and so is <math>\triangle{ABC}</math>, we find that <math>\triangle{ABC} \sim \triangle{AFE}</math>. Because <math>AB = \frac{AF}{2}</math> by symmetry across <math>BE</math>, we know that the similarity ratio is <math>\frac{1}{2}</math>.
   Thus the side length of <math>\triangle{AFE}</math> is <math>2</math>, and <math>FX = \frac{FE}{2} = 1</math>.
-  \\
@@ Line 214: / Line 214: @@
- \\
-  Recalling why we were doing all this, we find the area of <math>\triangle{FDX}</math>. It is <math>\frac{FX \times DX}{2} = \frac{1 \times \frac{\sqrt{3}}{2}}{2} = \boxed{\frac{\sqrt{3}}{4}}</math>. \\
+  Recalling why we were doing all this, we find the area of <math>\triangle{FDX}</math>. It is <math>\frac{FX \times DX}{2} = \frac{1 \times \frac{\sqrt{3}}{2}}{2} = \boxed{\frac{\sqrt{3}}{4}}</math>.
- \\
-  Let's summarize: \\
+  Let's summarize:
   \begin{center}

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

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