91Ó°ÊÓ

\(\left(-a^{2}+b^{2}+c^{2}\right) \tan A=\left(a^{2}-b^{2}+c^{2}\right) \tan B=\left(a^{2}+b^{2}-c^{2}\right) \tan C .\)

If \(A=30^{\circ}, a=7, b=8\) in \(\Delta A B C\), then how many values of \(B\) are possible?

The sides of a triangle are in A.P. and the greatest angle exceeds the least by \(90^{\circ}\); prove that the sides are proportional to \(\sqrt{7}+1, \sqrt{7}\) and \(\sqrt{7}-1\).

Two straight roads intersect at an angle of \(60^{\circ} .\) A bus on one road is \(2 \mathrm{~km}\). away from the intersection and a car on the other is \(3 \mathrm{~km}\). away from the intersection. Find the direct distance between the two vehicles

The side of a base of a square pyramid is \(a\) meters and it's vertex is at a height of \(h\) meters above the center of the base. If \(\theta \& \phi\) be respectively the inclinations of any face to the base and of any two faces to one another, prove that \(\tan \theta=\frac{2 h}{a}\) and \(\cot \frac{\phi}{2}=\sqrt{1+\frac{a^{2}}{2 h^{2}}}\).

The sides of a right angled triangle are 21 and \(28 \mathrm{~cm}\).; find the length of the perpendicular drawn to the hypotenuse from the right angle.

In the ambiguous case of the triangle, prove that the circumradius of the two triangles are equal.

\(D E F\) is the triangle formed by joining the points of contact of the incircle with the sides of the triangle \(A B C\); prove that:- i. it's sides are \(2 r \cos \frac{A}{2}, 2 r \cos \frac{B}{2}, 2 r \cos \frac{C}{2}\) ii. it's angles are \(\frac{\pi}{2}-\frac{A}{2}, \frac{\pi}{2}-\frac{B}{2}, \frac{\pi}{2}-\frac{C}{2}\) iii. it's area is \(\frac{2 \Delta^{3}}{a b c s}\), i.e. \(\frac{r \Delta}{2 R}\).

Let \(A B C\) be a triangle and let \(B B_{1}, C C_{1}\) be respectively the bisectors of \(\angle B, \angle C\) with \(B_{1}\) on \(A C\) and \(C_{1}\) on \(A B\). Let \(E, F\) be the feet of perpendiculars drawn from \(A\) on \(B B_{1}, C C_{1}\) respectively. Suppose \(D\) is the point at which the incircle of \(A B C\) touches \(A B\). Prove that \(A D=E F\).

PROVING IDENTITIES RELATED TO EX-RADII $$ a\left(r r_{1}+r_{2} r_{3}\right)=b\left(r r_{2}+r_{3} r_{1}\right)=c\left(r r_{3}+r_{1} r_{2}\right) $$