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 an acute angled triangle ABC, r + r_(1) = r_(2) + r_(3) and angleB gt (pi)/(3) , then

Incircle of A B C touches the sides BC, CA and AB at D, E and F, respectively. Let r_1 be the radius of incircle of B D Fdot Then prove that r_1=1/2((s-b)sinB)/((1+sin(B/2)))

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

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)

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)

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

prove that : triangle ABC, 1/r_1+1/r_2+1/r_3=1/r

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)

In a triangle ABC, prove that r^2+r_1^2+r_2^2+r_3^2=16R^2-a^2-b^2-c^2.

A circumcircle is a circle which passes through all vertices of a triangle and an incircle is a circle which is inscribed in a triangle touching all sides of a triangle. Let ABC be a right angled triangle whose radius of circumcircle is 5 and its one side AB = 6. The radius of incircle of triangle ABC is r. The value of r is