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(b + c)/p = (c + a)/q = (a + b)/r: find the sides, then the area, cosines or sines

This type of question, asked 4 times: 9 May 2024 S2, 13 May 2024 S2, 19 Apr 2025 S1, 19 May 2026 S1. All 4 questions below, oldest first.

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  1. Question 1

    9 May 2024 S2

    106
    Let a,b,ca, b, c be the lengths of sides of triangle ABC such that a+b7=b+c8=c+a9=k\frac{a+b}{7} = \frac{b+c}{8} = \frac{c+a}{9} = k. Then (A(△ABC))2k4=\dfrac{(A(\triangle ABC))^2}{k^4} =

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  2. Question 2

    13 May 2024 S2

    110
    In △ABC\triangle ABC, with notations, if b+c11=c+a12=a+b13\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}, then the value of cos⁡A+cos⁡B+cos⁡C\cos A+\cos B+\cos C is:

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  3. Question 3

    19 Apr 2025 S1

    150
    In a triangle ABC , with usual notations, if b+c11=c+a12=a+b13\frac{b + c}{11}=\frac{c + a}{12}=\frac{a + b}{13} Then cos⁡A:cos⁡B:cos⁡C\cos A:\cos B:\cos C is

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  4. Question 4

    19 May 2026 S1

    56
    In ΔABC\Delta ABC, with the usual notations, if a+b7=b+c8=c+a6=1\dfrac{a + b}{7} = \dfrac{b + c}{8} = \dfrac{c + a}{6} = 1, then sin⁡Asin⁡B=…\dfrac{\sin A}{\sin B} = \ldots

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