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Difference between revisions of "2023 WSMO Speed Round Problems/Problem 8"

(Created page with "==Problem== In regular octagon <math>ABCDEFGH</math> of sidelength <math>4,</math> quadrilaterals <math>ACEG</math> and <math>BDFH</math> are drawn. Find the square of the ar...")
 
Line 4: Line 4:
  
 
==Solution==
 
==Solution==
 +
<asy>
 +
pair a = dir(112.5);
 +
pair b = dir(67.5);
 +
pair c = dir(22.5);
 +
pair d = dir(337.5);
 +
pair e = dir(292.5);
 +
pair f = dir(247.5);
 +
pair g = dir(202.5);
 +
pair h = dir(157.5);
 +
 +
label("$A$", a, NW);
 +
label("$B$", b, NE);
 +
label("$C$", c, E);
 +
label("$D$", d, E);
 +
label("$E$", e, SE);
 +
label("$F$", f, SW);
 +
label("$G$", g, W);
 +
label("$H$", h, W);
 +
 +
draw(a--b--c--d--e--f--g--h--cycle,black+linewidth(2));
 +
draw(a--c--e--g--cycle,blue+linewidth(1));
 +
draw(b--d--f--h--cycle,blue+linewidth(1));
 +
 +
pair X = intersectionpoint(b--h,a--c);
 +
dot(X);
 +
label("$X$", X, S);
 +
 +
pair Y = intersectionpoint(b--h,a--g);
 +
dot(Y);
 +
label("$Y$", Y, SE);
 +
</asy>
 +
 +
Let <math>X = BH\cap AC</math> and <math>Y = BH\cap AG</math>.

Revision as of 11:43, 12 September 2025

Problem

In regular octagon $ABCDEFGH$ of sidelength $4,$ quadrilaterals $ACEG$ and $BDFH$ are drawn. Find the square of the area of the overlap of the two quadrilaterals.

Solution

[asy] pair a = dir(112.5); pair b = dir(67.5); pair c = dir(22.5); pair d = dir(337.5); pair e = dir(292.5); pair f = dir(247.5); pair g = dir(202.5); pair h = dir(157.5); label("$A$", a, NW); label("$B$", b, NE); label("$C$", c, E); label("$D$", d, E); label("$E$", e, SE); label("$F$", f, SW); label("$G$", g, W); label("$H$", h, W); draw(a--b--c--d--e--f--g--h--cycle,black+linewidth(2)); draw(a--c--e--g--cycle,blue+linewidth(1)); draw(b--d--f--h--cycle,blue+linewidth(1)); pair X = intersectionpoint(b--h,a--c); dot(X); label("$X$", X, S); pair Y = intersectionpoint(b--h,a--g); dot(Y); label("$Y$", Y, SE); [/asy]

Let $X = BH\cap AC$ and $Y = BH\cap AG$.

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