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Equilateral triangle with one side along a given line

This type of question, asked 4 times: 2021 Paper 10, 26 Jun 2022, 28 Jan 2026 S1, 5 Apr 2026 S1. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 10

    65
    Let the centroid of an equilateral triangle ABCABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x+y=3x+y= 3. If RR and rr be the radius of circumcircle and incircle respectively of △ABC\bigtriangleup ABC, then (R+r)(R + r) is equal to:

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  2. Question 2

    26 Jun 2022

    71
    Let RR be the point (3,7)(3,7) and let PP and QQ be two points on the line x+y=5x + y = 5 such that PQRPQR is an equilateral triangle. Then the area of △PQR\bigtriangleup PQR is:

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  3. Question 3

    28 Jan 2026 S1

    56
    Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line x+22y=4x+ 2\sqrt{2}y= 4. If the co-ordinates of the vertex A are (α,β)(\alpha,\beta), then the greatest integer less than or equal to ∣α+2β∣|\alpha+\sqrt{2}\beta| is

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  4. Question 4

    5 Apr 2026 S1

    61
    In an equilateral triangle PQR , let the vertex P be at (3,5)(3,5) and the side QR be along the line x+y=4x+y= 4. If the orthocenter of the triangle PQRPQR is (α,β)(\alpha,\beta), then 9(α+β)9(\alpha+\beta) is equal to :

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