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Omni-MATH / Let ABC be an equilateral triangle with side length 1. Points D,E,F lie inside triangle ABC such that A,E,F are collinear, B,F,D are collinear, C,D,E are collinear, and triangle …

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problem

Let ABCA B C be an equilateral triangle with side length 1. Points D,E,FD, E, F lie inside triangle ABCA B C such that A,E,FA, E, F are collinear, B,F,DB, F, D are collinear, C,D,EC, D, E are collinear, and triangle DEFD E F is equilateral. Suppose that there exists a unique equilateral triangle XYZX Y Z with XX on side BC¯,Y\overline{B C}, Y on side AB¯\overline{A B}, and ZZ on side AC¯\overline{A C} such that DD lies on side XZ¯,E\overline{X Z}, E lies on side YZ¯\overline{Y Z}, and FF lies on side XY¯\overline{X Y}. Compute AZA Z.
Plain-text mathematical notation (without MathML)
Let ABC be an equilateral triangle with side length 1. Points D,E,F lie inside triangle ABC such that A,E,F are collinear, B,F,D are collinear, C,D,E are collinear, and triangle DEF is equilateral. Suppose that there exists a unique equilateral triangle XYZ with X on side (BC)¯,Y on side (AB)¯, and Z on side (AC)¯ such that D lies on side (XZ)¯,E lies on side (YZ)¯, and F lies on side (XY)¯. Compute AZ.
Original LaTeX notation
Let $A B C$ be an equilateral triangle with side length 1. Points $D, E, F$ lie inside triangle $A B C$ such that $A, E, F$ are collinear, $B, F, D$ are collinear, $C, D, E$ are collinear, and triangle $D E F$ is equilateral. Suppose that there exists a unique equilateral triangle $X Y Z$ with $X$ on side $\overline{B C}, Y$ on side $\overline{A B}$, and $Z$ on side $\overline{A C}$ such that $D$ lies on side $\overline{X Z}, E$ lies on side $\overline{Y Z}$, and $F$ lies on side $\overline{X Y}$. Compute $A Z$.

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Source and history

task_id
f48a7d89-9de1-5f6d-a827-45a266a16439
task_key
test--f48a7d89-9de1-5f6d-a827-45a266a16439
task_revision_id
1
config / split
/ test

Official source

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