Home
Class 12
MATHS
Prove that r(1)^(2)+r(2)^(2) +r(3)^(3)...

Prove that
`r_(1)^(2)+r_(2)^(2) +r_(3)^(3) +r^(2) =16R^(2) -a^(2) -b^(2) -c^(2).`
where r= in radius, R = circumradius,, `r_(1), r_(2), r_(3)` are ex-radii.

Text Solution

Verified by Experts

The correct Answer is:
`=16R^(2) -a^(2)-b^(2)-c^(2)=RHS`
Promotional Banner

Topper's Solved these Questions

  • PROPERTIES AND SOLUTION OF TRIANGLES

    ARIHANT MATHS|Exercise Exercise For Sesssion 1|20 Videos

  • PROPERTIES AND SOLUTION OF TRIANGLES

    ARIHANT MATHS|Exercise Exercise For Sesssion 2|19 Videos

  • PRODUCT OF VECTORS

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|52 Videos

  • SEQUENCES AND SERIES

    ARIHANT MATHS|Exercise Exercise (Questions Asked In Previous 13 Years Exam)|38 Videos

Similar Questions

Explore conceptually related problems

Prove that r_(1)+r_(2)+r_(3)-r=4R

Prove that r_(1)+r_(2)+r_(3)-r=4R

Show that 16R^(2)r r_(1) r _(2) r_(3)=a^(2) b ^(2) c ^(2)

Prove that r_(1) r_(2) + r_(2) r_(3) + r_(3) r_(1) = (1)/(4) (a + b + c)^(2)

Prove that : (r_1-r)/(a) +(r_2-r)/(b) = (c )/(r_3) .

Show that (r_(1)+ r _(2))(r _(2)+ r _(3)) (r_(3)+r_(1))=4Rs^(2)

Prove that (r_(1)+r_(2))/(1+cos C)=2R

Prove that : (r_1)/(b c)+(r_2)/(c a)+(r_3)/(a b)=1/r-1/(2R)

Prove that (r_(1) -r)/(a) + (r_(2) -r)/(b) = (c)/(r_(3))

show that r_(2)r_(3) +r_(3) r_(1)+ r _(1) r_(2)=s^(2)

ARIHANT MATHS-PROPERTIES AND SOLUTION OF TRIANGLES -Exercise (Questions Asked In Previous 13 Years Exam)
  1. Prove that r(1)^(2)+r(2)^(2) +r(3)^(3) +r^(2) =16R^(2) -a^(2) -b^(2)...

    Text Solution

    |

  2. In a triangle XYZ, let x, y, z be the lengths of sides opposite to the...

    Text Solution

    |

  3. In a triangle the sum of two sides is x and the product of the same is...

    Text Solution

    |

  4. Consider a triangle A B C and let a , ba n dc denote the lengths of th...

    Text Solution

    |

  5. In a triangle PQR, P is the largest angle and cosP=1/3. Further the in...

    Text Solution

    |

  6. Let PQR be a triangle of area Delta with a = 2, b = 7/2 and c =5/2, wh...

    Text Solution

    |

  7. If the angle A ,Ba n dC of a triangle are in an arithmetic propression...

    Text Solution

    |

  8. Let A B C be a triangle such that /A C B=pi/6 and let a , ba n dc deno...

    Text Solution

    |

  9. a triangle A B C with fixed base B C , the vertex A moves such that co...

    Text Solution

    |

  10. Let A B Ca n dA B C ' be two non-congruent triangles with sides A B=4,...

    Text Solution

    |

  11. A straight line through the vertex P of a triangle PQR intersects the ...

    Text Solution

    |

  12. Consider the circle x^2 + y^2 = 9 and the parabola y^2 = 8x. They inte...

    Text Solution

    |

  13. Consider the circle x^2 + y^2 = 9 and the parabola y^2 = 8x. They inte...

    Text Solution

    |

  14. Consider the circle x^2 + y^2 = 9 and the parabola y^2 = 8x. They inte...

    Text Solution

    |

  15. Internal bisector of /A of triangle ABC meets side BC at D. A line dra...

    Text Solution

    |

  16. Given an isosceles triangle, whose one angle is-2pi/3 and the radius o...

    Text Solution

    |

  17. In Delta ABC, which one is true among the following ?

    Text Solution

    |

  18. Let a vertical tower A b have its end A on the level ground. Let C be ...

    Text Solution

    |

  19. ABCD is a trapezium such that AB and CD are parallel and BC | CD. If ...

    Text Solution

    |

  20. For a regular polygon, let r and R be the radii of the inscribed and ...

    Text Solution

    |

  21. In triangle A B C , let /c=pi/2dot If r is the inradius and R is circu...

    Text Solution

    |