benchmarks.wiki

benchmarks.wiki / Public workspace

Omni-MATH / For a point P=(a,a²) in the coordinate plane, let ℓ(P) denote the line passing through P with slope 2a . Consider the set of triangles with vertices of the form P₁=(a₁,a₁²) , …

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

For a point P=(a,a2)P = (a, a^2) in the coordinate plane, let ℓ(P)\ell(P) denote the line passing through PP with slope 2a2a . Consider the set of triangles with vertices of the form P1=(a1,a12)P_1 = (a_1, a_1^2) , P2=(a2,a22)P_2 = (a_2, a_2^2) , P3=(a3,a32)P_3 = (a_3, a_3^2) , such that the intersections of the lines ℓ(P1)\ell(P_1) , ℓ(P2)\ell(P_2) , ℓ(P3)\ell(P_3) form an equilateral triangle △\triangle . Find the locus of the center of △\triangle as P1P2P3P_1P_2P_3 ranges over all such triangles.
Plain-text mathematical notation (without MathML)
For a point P=(a,a²) in the coordinate plane, let ℓ(P) denote the line passing through P with slope 2a . Consider the set of triangles with vertices of the form P₁=(a₁,a₁²) , P₂=(a₂,a₂²) , P₃=(a₃,a₃²) , such that the intersections of the lines ℓ(P₁) , ℓ(P₂) , ℓ(P₃) form an equilateral triangle △ . Find the locus of the center of △ as P₁P₂P₃ ranges over all such triangles.
Original LaTeX notation
For a point $P = (a, a^2)$ in the coordinate plane, let $\ell(P)$ denote the line passing through $P$ with slope $2a$ . Consider the set of triangles with vertices of the form $P_1 = (a_1,  a_1^2)$ , $P_2 = (a_2, a_2^2)$ , $P_3 = (a_3, a_3^2)$ , such that the intersections of the lines $\ell(P_1)$ , $\ell(P_2)$ , $\ell(P_3)$ form an equilateral triangle $\triangle$ . Find the locus of the center of $\triangle$ as $P_1P_2P_3$ ranges over all such triangles.

Discussion

Discussion

No discussion posts on this page yet. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. Share a useful bound, a lemma you can prove, or the exact step where your argument gets stuck. Use the posting template.

Source and history

task_id
c10c2865-34ea-5df3-9570-e8354a9dd363
task_key
test--c10c2865-34ea-5df3-9570-e8354a9dd363
task_revision_id
1
config / split
/ test

Official source

initial import