`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 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))

In an acute angle triange ABC, a semicircle with radius r_(a) is constructed with its base on BC and tangent to the other two sides r_(b) and r_(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))

Let A B C be a triangle with incenter I and inradius rdot Let D ,E ,a n dF be the feet of the perpendiculars from I to the sides B C ,C A ,a n dA B , respectively. If r_1,r_2a n dr_3 are the radii of circles inscribed in the quadrilaterals A F I E ,B D I F ,a n dC E I D , respectively, prove that (r_1)/(r-1_1)+(r_2)/(r-r_2)+(r_3)/(r-r_3)=(r_1r_2r_3)/((r-r_1)(r-r_2)(r-r_3))

In triangle A B C , let /_c=pi/2dot If r is the inradius and R is circumradius of the triangle, then 2(r+R) is equal to (a) a+b (b) b+c (c) c+a (d) a+b+c

In triangle A B C , let /_c=pi/2dot If r is the inradius and R is circumradius of the triangle, then 2(r+R) is equal to a+b (b) b+c c+a (d) a+b+c

O P Q R is a square and M ,N are the middle points of the sides P Qa n dQ R , respectively. Then the ratio of the area of the square to that of triangle O M N is (a)4:1 (b) 2:1 (c) 8:3 (d) 7:3

If R_1 is the circumradius of the pedal triangle of a given triangle A B C ,a n dR_2 is the circumradius of the pedal triangle of the pedal triangle formed, and so on R_3,R_4 then the value of sum_(i=1)^ooR_i , where R (circumradius) of A B C is 5 is 8 (b) 10 (c) 12 (d) 15

Prove that ("a c o s"A+b cosB+ccosC)/(a+b+c)=r/Rdot

Prove that ("a c o s"A+b cosB+ccosC)/(a+b+c)=r/Rdot