`O` is the circumcenter of ` A B Ca n dR_1, R_2, R_3` are respectively, the radii of the circumcircles of the triangle `O B C ,O C A` and OAB. Prove that `a/(R_1)+b/(R_2)+c/(R_3),(a b c)/(R_3)`

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)

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

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

In an acute angled triangle A B C , a semicircle with radius r_a is constructed with its base on BC and tangent to the other two sides. r_b a n dr_c are defined similarly. If r is the radius of the incircle of triangle ABC then prove that 2/r=1/(r_a)+1/(r_b)+1/(r_c)dot

In a triangle ABC prove that a(rr_1+r_2r_3)=b(rr_2+r_3r_1)=c(rr_3+r_1r_2)=abc

O is any point on the diagonal B D of the parallelogram A B C D . Prove that a r( /_\O A B)=a r( /_\O B C)

The diagonals of quadrilateral A B C D intersect at O . Prove that (a r( A C B))/(a r( A C D))=(B O)/(D O)

Three resistors R_1, R_2 and R_3 are to be combined to form an electrical circuit as shown in the figure. It is found that when R_1 R_2 and R_3 are put respectively in positions A, B and C, the effective resistance of the circuit is 70 Omega When R_2, R_3 and R_1 are put respectively in position A, B and C the effective resistance is 35 Omega and when R_3, R_1 and R_2 are respectively put in the position A, B and C, the effective resistance is 42 Omega The resistance R_(1): R_(2): R_(3 ) are in the ratio

Consider a D E F , the pedal triangle of the A B C such that A-F-B\ a n d\ \ B-D-C are collinear. If H is the incentre of D E F and R_1,\ R_2,\ R_3 are the circumradii of the quadarilaterals A F H E ; B D H F\ a n d\ C E H D respectively, then prove that sumR_1=R+r where R is the circumradius and r is the inradius of A B Cdot