In triangle `A B C ,\ ` the est values of `sumcos e c(A/2)a n d\ sumsec^2(A/2)` are 6 and 4 respectively.

If a,b,and c are the side of a triangle and A,B and C are the angles opposite to a,b,and c respectively, then Delta=|{:(a^(2),bsinA,CsinA),(bsinA,1,cosA),(CsinA,cos A,1):}| is independent of

Consider a Delta ABC and let a,b, and c denote the leghts of the sides opposite to vertices A,B and C, respectively. Suppose a=2,b =3, c=4 and H be the orthocentre. Find 15(HA)^(2).

The vertices of a Delta ABC are A(4,6), B(1,5) and C(7,2) . A line is drawn to intersect sides AB and AC at D and E respectively, such that (AD)/(AB) = (AE)/(AC) = (1)/(4) . Calculate the area of the Delta ADE and compare it with the area of Delta ABC . (Recall Theorem 6 . 2 and Theorem 6 . 6)

In triangle A B C , if cotA ,cotB ,cotC are in AdotPdot, then a^2,b^2,c^2 are in ____________ progression.

Statement 1: If vec u a n d vec v are unit vectors inclined at an angle alphaa n d vec x is a unit vector bisecting the angle between them, then vec x=( vec u+ vec v)//(2sin(alpha//2)dot Statement 2: If "Delta"A B C is an isosceles triangle with A B=A C=1, then the vector representing the bisector of angel A is given by vec A D=( vec A B+ vec A C)//2.

In an acute triangle A B C if sides a , b are constants and the base angles Aa n dB vary, then show that (d A)/(sqrt(a^2-b^2sin^2A))=(d B)/(sqrt(b^2-a^2sin^2B))

For any triangle ABC, prove that : (a+b)/(c)=(cos((A-B)/(2)))/(sin""(C)/(2))

If a variable tangent to the curve x^2y=c^3 makes intercepts a , bonx-a n dy-a x e s , respectively, then the value of a^2b is 27c^3 (b) 4/(27)c^3 (c) (27)/4c^3 (d) 4/9c^3

For any triangle ABC, prove that : (b+c)cos((B+C)/(2))=acos((B-C)/(2))