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Homefsc 1st year math solutionsSolution of circles connected with triangles Exercise 12.8

Solution of circles connected with triangles Exercise 12.8

  1. Show that: i) r=4 R \sin \frac{\alpha}{2} \sin \frac{\beta}{2} \sin \frac{\gamma}{2}
    ii) s=4 R \cos \frac{\alpha}{2} \cos \frac{\beta}{2} \cos \frac{\gamma}{2}
  2. Show that: r=a \sin \frac{\beta}{2} \sin \frac{\gamma}{2} \sec \frac{\alpha}{2}=b \sin \frac{\gamma}{2} \sin \frac{\alpha}{2} \sec \frac{\beta}{2}=c \sin \frac{\alpha}{2} \sin \frac{\beta}{2} \sec \frac{\gamma}{2}
  3. Show that: i) r_1=4 R \sin \frac{\alpha}{2} \cos \frac{\beta}{2} \cos \frac{\gamma}{2}
    ii) r_2=4 R \cos \frac{\alpha}{2} \sin \frac{\beta}{2} \cos \frac{\gamma}{2}
    iii) r_3=4 R \cos \frac{\alpha}{2} \cos \frac{\beta}{2} \sin \frac{\gamma}{2}
  4. Show that:
    i) r_1=s \tan \frac{\alpha}{2}
    ii) r_2=s \tan \frac{\beta}{2}
    iii) r_3=s \tan \frac{\gamma}{2}
  5. Prove that:
    i) r_1 r_2+r_2 r_3+r_3 r_1=s^2
    ii) r r_1 r_2 r_3=\Delta^2
    iii) r_1+r_2+r_3-r=4 R
    iv) r_1 r_2 r_3=r s^2
  6. Find R, r_1 r_1, r_2 and r_3, if measures of the sides of triangle A B C are
    i) a=13, b=14, c=15
    ii) a=34, b=20, c=42
  7. Prove that in an equilateral triangle,
    i) r: R: r_1=1: 2: 3
    ii) r: R: r_1: r_2: r_3=1: 2: 3: 3: 3
  8. Prove that:
    i) \Delta=r^2 \cot \frac{\alpha}{2} \cot \frac{\beta}{2} \cot \frac{\gamma}{2}
    ii) r=s \tan \frac{\alpha}{2} \tan \frac{\beta}{2} \tan \frac{\gamma}{2}
    iii) \Delta=4 R r \cos \frac{\alpha}{2} \cos \frac{\beta}{2} \cos \frac{\gamma}{2}
  9. Show that: i) \frac{1}{2 r R}=\frac{1}{a b}+\frac{1}{b c}+\frac{1}{c a}
    ii) \frac{1}{r}=\frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}
  10. Prove that:

        \[r=\frac{a \sin \frac{\beta}{2} \sin \frac{\gamma}{2}}{\cos \frac{\alpha}{2}}=\frac{b \sin \frac{\alpha}{2} \cdot \sin \frac{\gamma}{2}}{\cos \frac{\beta}{2}}=\frac{c \sin \frac{\alpha}{2} \cdot \sin \frac{\beta}{2}}{\cos \frac{\gamma}{2}}\]

  11. Prove that: a b c(\sin \alpha+\sin \beta+\sin \gamma)=4 \Delta s.
  12. Prove that: i) \left(r_1+r_2\right) \tan \frac{\gamma}{2}=c.
    ii) \left(r_3-r\right) \cot \frac{\gamma}{2}=c
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