If `I` is the incenter of ` A B Ca n dR_1, R_2,a n dR_3` re, respectively, the radii of the circumcircles of the triangles `I B C ,I C Aa n dI A B ,` then prove that `R_1R_2R_3=2r R^2dot`

If I is the incenter of Delta ABC and R_(1), R_(2), and R_(3) are, respectively, the radii of the circumcircle of the triangle IBC, ICA, and IAB, then prove that R_(1) R_(2) R_(3) = 2r R^(2)

O is the circumcenter of bigoplus ABC and R_(1),R_(2),R_(3) are respectively,the radii of the circumcircles of the triangle OBC,OCA and OAB.Prove that (a)/(R_(1))+(b)/(R_(2))+(c)/(R_(3)),(abc)/(R_(3))

If ' O ' is the circum centre of the A B C and R_1,\ R_2,\ R_3 are the radii of the circumcircle of triangle O B C ,\ O C A\ a n d\ O A B respectively then a/(R_1)+b/(R_2)+c/(R_3) has the value equal to (a b c)/(2R^3) (b) (R^3)/(a b c) (c) (4)/(R^2) (d) (a b c)/(R^3)

O is the circumcentre of the triangle ABC and R_(1),R_(2),R_(3) are the radii of the circumcirclesof the triangles OBC,OCA and OAB respectively,then (a)/(R_(1))+(b)/(R_(2))+(c)/(R_(3)), is equal to

In a right angled DeltaABC, angleA=90^(@) . If AC = b, BC = a, AB = c and r and R are respectively radii of incircle and circumcircle of the triangle then prove that 2(r + R) = b + c

In A A B C ,\ P\ a n d\ Q are respectively the mid-points of A B\ a n d\ B C and R is the mid-point of A Pdot Prove that: a r\ (\ P R Q)=1/2a r\ (\ A R C)

In any A B C , prove that r^2+r1 2+r2 2+r3 2=16 R^2-a^2-b^2-c^2dot

In A A B C ,\ P\ a n d\ Q are respectively the mid-points of A B\ a n d\ B C and R is the mid-point of A Pdot Prove that: a r\ ( R Q C)=3/8\ a r\ (\ A B C)

Let A B C be a triangle right-angled at Aa n dS be its circumcircle. Let S_1 be the circle touching the lines A B and A C and the circle S internally. Further, let S_2 be the circle touching the lines A B and A C produced and the circle S externally. If r_1 and r_2 are the radii of the circles S_1 and S_2 , respectively, show that r_1r_2=4 area ( A B C)dot