If `I_1, I_2, I_3` are the centers of escribed circles of triangle ` A B C ,` show that area of triangle ` I_1I_2I_3=(a b c)/(2r)dot`

In triangle A B C ,/_A=60^0,/_B=40^0,a n d/_C=80^0dot If P is the center of the circumcircle of triangle A B C with radius unity, then the radius of the circumcircle of triangle B P C is (a) 1 (b) sqrt(3) (c) 2 (d) sqrt(3) 2

If z_1, z_2 and z_3 , are the vertices of an equilateral triangle ABC such that |z_1 -i| = |z_2 -i| = |z_3 -i| .then |z_1 +z_2+ z_3| equals:

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)

If two arms of any triangle are given by two vectors veca and vecb , then show that the area of the triangle = 1/2 I veca xx vecb I.

If R_(1) is the circumradius of the pedal triangle of a given triangle ABC, and R_(2) is the circumradius of the pedal triangle of the pedal triangle formed, and so on R_(3), R_(4) ..., then the value of underset( i=1)overset(oo)sum R_(i) , where R (circumradius) of DeltaABC is 5 is

Let O be the circumcenter, H be the orthocentre, I be the incenter, and I_1, I_2,I_3 be the excenters of acute-angled A B Cdot Column I Column II Angle subtended by O I at vertex A p. |B-C| Angle subtended by H I at vertex A q. |(B-C)/2| Angle subtended by O H at vertex A r. (B+C)/2 Angle subtended by I_2I_3a tI_1 A q. B/2-C

If A,B,C are the angles of a triangle, show that, (i) sin B cos(C +A) + cos B sin (C +A) = 0

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))

A(z_1),B(z_2),C(z_3) are the vertices of the triangle A B C (in anticlockwise). If /_A B C=pi//4 and A B=sqrt(2)(B C) , then prove that z_2=z_3+i(z_1-z_3)dot

In triangle ABC, (i) asin(A/2 + B) = (b+c) sin A/2