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Difference between revisions of "2006 USAMO Problems/Problem 6"

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From the similarity, we have that <math>XE/XF=AE/BF</math>. But we are given <math>ED/AE=CF/BF</math>, so multiplying the 2 equations together gets us <math>ED/FC=XE/XF</math>. <math>DEX,CFX</math> are the supplements of <math>AEX, BFX</math>, which are congruent, so <math>DEX=CFX</math>, and so <math>XED\sim XFC</math> by SAS similarity, and so <math>X</math> is also the center of spiral similarity for <math>E,D,F,</math> and <math>C</math>. Thus, <math>X</math> and <math>Y</math> are the same point, which all the circumcircles pass through, and so the statement is true.
 
From the similarity, we have that <math>XE/XF=AE/BF</math>. But we are given <math>ED/AE=CF/BF</math>, so multiplying the 2 equations together gets us <math>ED/FC=XE/XF</math>. <math>DEX,CFX</math> are the supplements of <math>AEX, BFX</math>, which are congruent, so <math>DEX=CFX</math>, and so <math>XED\sim XFC</math> by SAS similarity, and so <math>X</math> is also the center of spiral similarity for <math>E,D,F,</math> and <math>C</math>. Thus, <math>X</math> and <math>Y</math> are the same point, which all the circumcircles pass through, and so the statement is true.
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=== Solution 2 ===
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We will give a solution using complex coordinates. The first step is the following lemma.
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'''Lemma.''' Suppose <math>s</math> and <math>t</math> are real numbers and <math>x</math>, <math>y</math> and <math>z</math> are complex. The circle in the complex plane passing through <math>x</math>, <math>x + ty</math> and <math>x + (s + t)z</math> also passes through the point <math>x + syz/(y - z)</math>, independent of <math>t</math>.
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''Proof.'' Four points <math>z_1</math>, <math>z_2</math>, <math>z_3</math> and <math>z_4</math> in the complex plane lie on a circle if and only if the cross-ratio
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<cmath>cr(z_1, z_2, z_3, z_4) = \frac{(z_1 - z_3)(z_2 - z_4)}{(z_1 - z_4)(z_2 - z_3)}</cmath>
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is real. Since we compute
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<cmath>cr(x, x + ty, x + (s + t)z, x + syz/(y - z)) = \frac{s + t}{s}</cmath>
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the given points are on a circle. <math>\blacksquare</math>
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Lay down complex coordinates with <math>S = 0</math> and <math>E</math> and <math>F</math> on the positive real axis. Then there are real <math>r_1</math>, <math>r_2</math> and <math>R</math> with <math>B = r_1A</math>, <math>F = r_2E</math> and <math>D = E + R(A - E)</math> and hence <math>AE/ED = BF/FC</math> gives
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<cmath>C = F + R(B - F) = r_2(1 - R)E + r_1RA.</cmath>
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The line <math>CD</math> consists of all points of the form <math>sC + (1 - s)D</math> for real <math>s</math>. Since <math>T</math> lies on this line and has zero imaginary part, we see from <math>\text{Im}(sC + (1 - s)D) = (sr_1R + (1 - s)R)\text{Im}(A)</math> that it corresponds to <math>s = -1/(r_1 - 1)</math>. Thus
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<cmath>T = \frac{r_1D - C}{r_1 - 1} = \frac{(r_2 - r_1)(R - 1)E}{r_1 - 1}.</cmath>
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Apply the lemma with <math>x = E</math>, <math>y = A - E</math>, <math>z = (r_2 - r_1)E/(r_1 - 1)</math>, and <math>s = (r_2 - 1)(r_1 - r_2)</math>. Setting <math>t = 1</math> gives
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<cmath>(x, x + y, x + (s + 1)z) = (E, A, S = 0)</cmath>
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and setting <math>t = R</math> gives
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<cmath>(x, x + Ry, x + (s + R)z) = (E, D, T).</cmath>
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Therefore the circumcircles to <math>SAE</math> and <math>TDE</math> meet at
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<cmath>x + \frac{syz}{y - z} = \frac{AE(r_1 - r_2)}{(1 - r_1)E - (1 - r_2)A} = \frac{AF - BE}{A + F - B - E}.</cmath>
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This last expression is invariant under simultaneously interchanging <math>A</math> and <math>B</math> and interchanging <math>E</math> and <math>F</math>. Therefore it is also the intersection of the circumcircles of <math>SBF</math> and <math>TCF</math>.
  
 
{{alternate solutions}}
 
{{alternate solutions}}

Revision as of 08:16, 6 August 2014

Problem

(Zuming Feng, Zhonghao Ye) Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$, respectively, such that $AE/ED = BF/FC$. Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$ respectively. Prove that the circumcircles of triangles $SAE$, $SBF$, $TCF$, and $TDE$ pass through a common point.

Solutions

Solution 1

Let the intersection of the circumcircles of $SAE$ and $SBF$ be $X$, and let the intersection of the circumcircles of $TCF$ and $TDE$ be $Y$.

$BXF=BSF=AXE$ because $BSF$ tends both arcs $AE$ and $BF$. $BFX=XSB=XEA$ because $XSB$ tends both arcs $XA$ and $XB$. Thus, $XAE\sim XBF$ by AA similarity, and $X$ is the center of spiral similarity for $A,E,B,$ and $F$. $FYC=FTC=EYD$ because $FTC$ tends both arcs $ED$ and $FC$. $FCY=FTY=EDY$ because $FTY$ tends both arcs $YF$ and $YE$. Thus, $YED\sim YFC$ by AA similarity, and $Y$ is the center of spiral similarity for $E,D,F,$ and $C$.

From the similarity, we have that $XE/XF=AE/BF$. But we are given $ED/AE=CF/BF$, so multiplying the 2 equations together gets us $ED/FC=XE/XF$. $DEX,CFX$ are the supplements of $AEX, BFX$, which are congruent, so $DEX=CFX$, and so $XED\sim XFC$ by SAS similarity, and so $X$ is also the center of spiral similarity for $E,D,F,$ and $C$. Thus, $X$ and $Y$ are the same point, which all the circumcircles pass through, and so the statement is true.

Solution 2

We will give a solution using complex coordinates. The first step is the following lemma.

Lemma. Suppose $s$ and $t$ are real numbers and $x$, $y$ and $z$ are complex. The circle in the complex plane passing through $x$, $x + ty$ and $x + (s + t)z$ also passes through the point $x + syz/(y - z)$, independent of $t$.

Proof. Four points $z_1$, $z_2$, $z_3$ and $z_4$ in the complex plane lie on a circle if and only if the cross-ratio \[cr(z_1, z_2, z_3, z_4) = \frac{(z_1 - z_3)(z_2 - z_4)}{(z_1 - z_4)(z_2 - z_3)}\] is real. Since we compute \[cr(x, x + ty, x + (s + t)z, x + syz/(y - z)) = \frac{s + t}{s}\] the given points are on a circle. $\blacksquare$

Lay down complex coordinates with $S = 0$ and $E$ and $F$ on the positive real axis. Then there are real $r_1$, $r_2$ and $R$ with $B = r_1A$, $F = r_2E$ and $D = E + R(A - E)$ and hence $AE/ED = BF/FC$ gives \[C = F + R(B - F) = r_2(1 - R)E + r_1RA.\] The line $CD$ consists of all points of the form $sC + (1 - s)D$ for real $s$. Since $T$ lies on this line and has zero imaginary part, we see from $\text{Im}(sC + (1 - s)D) = (sr_1R + (1 - s)R)\text{Im}(A)$ that it corresponds to $s = -1/(r_1 - 1)$. Thus \[T = \frac{r_1D - C}{r_1 - 1} = \frac{(r_2 - r_1)(R - 1)E}{r_1 - 1}.\] Apply the lemma with $x = E$, $y = A - E$, $z = (r_2 - r_1)E/(r_1 - 1)$, and $s = (r_2 - 1)(r_1 - r_2)$. Setting $t = 1$ gives \[(x, x + y, x + (s + 1)z) = (E, A, S = 0)\] and setting $t = R$ gives \[(x, x + Ry, x + (s + R)z) = (E, D, T).\] Therefore the circumcircles to $SAE$ and $TDE$ meet at \[x + \frac{syz}{y - z} = \frac{AE(r_1 - r_2)}{(1 - r_1)E - (1 - r_2)A} = \frac{AF - BE}{A + F - B - E}.\] This last expression is invariant under simultaneously interchanging $A$ and $B$ and interchanging $E$ and $F$. Therefore it is also the intersection of the circumcircles of $SBF$ and $TCF$.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

See also

  • <url>viewtopic.php?t=84559 Discussion on AoPS/MathLinks</url>
2006 USAMO (Problems • Resources)
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