91Ó°ÊÓ

If \(D\) is mid-point of \(C A\) in triangle \(A B C\) and \(\Delta\) is the area of triangle, then prove that \(\tan (\angle A D B)=\frac{4 \Delta}{a^{2}-c^{2}}\)

Prove that the distance between the circum-centre ( \(O\) ) and the in-center \((I)\) is \(O I=R \times \sqrt{\left(1-8 \sin \left(\frac{A}{2}\right) \sin \left(\frac{B}{2}\right) \sin \left(\frac{C}{2}\right)\right)}\)

In any triangle \(A B C\), prove that, \(4 \Delta(\cot A+\cot B+\cot C)=a^{2}+b^{2}+c^{2}\)

Prove that the ratio of circum-radius and in-radius of 3 : an equilateral triangle is \(1 / 2\).

If the median of \(\Delta A B C\), through \(A\) is perpendicular to \(A B\), then (a) \(\tan A+\tan B=0\) (b) \(2 \tan A+\tan B=0\) (c) \(\tan A+2 \tan B=0\) (d) None

In a \(\Delta A B C, \cos A+\cos B+\cos C=\frac{3}{2}\), then the \(\Delta\) (a) Isosceles (b) right angled (c) equilateral (d) None

In any triangle \(A B C\), prove that, \(\left(\frac{2 a b c}{a+b+c}\right) \cdot \cos \left(\frac{A}{2}\right) \cos \left(\frac{B}{2}\right) \cos \left(\frac{C}{2}\right)=\Delta\)

Prove that the distance of the orthocenter from the sides and angular points of a triangle is \(2 R \cos A, 2 R \cos B\) and \(2 R \cos C .\)

If in a triangle \(A B C, a=6, b=3\) and \(\cos (A-B)=\frac{4}{5}\), then find its area.

If in a triangle \(A B C, \angle A=30^{\circ}\) and the area of the triangle is \(\frac{a^{2} \sqrt{3}}{4}\), then prove that either \(B=4 C\) or \(C=4 B\)