Chapter 7

If in a triangle \(A B C, \frac{\cos A+2 \cos C}{\cos A+2 c \cos B}=\frac{\sin B}{\sin C}\), prove

In \(\triangle A B C, a \geq b \geq c\), if \(\frac{a^{3}+b^{3}+c^{3}}{\sin ^{3} A+\sin ^{3} B+\sin ^{3} C}=8\), then the maximum value of \(a\) is (a) \(\frac{1}{2}\) (b) 2 (c) 8 (d) 64

In any triangle \(A B C\), prove that \(\left(b^{2}-c^{2}\right) \cot A+\left(c^{2}-a^{2}\right) \cot B+\left(a^{2}-b^{2}\right) \cot C=0\)

In a triangle \(A B C\), if \(a \tan A+b \tan B=(a+b) \tan \left(\frac{A+B}{2}\right)\) prove that the triangle is isosceles.

Sides of a triangle \(A B C\) are in A.P. If \(a<\min \\{b, c\\}\), then \(\cos A\) may be equal to (a) \(\frac{3 c-4 b}{2 a}\) (b) \(\frac{3 c-4 b}{2 c}\) (c) \(\frac{4 c-3 b}{2 b}\) (d) \(\frac{4 c-3 b}{2 c}\)

In a tringle \(\Delta A B C\), prove that \(a \sin (B-C)+b \sin (C-A)+c \sin (A-B)=0\)

In a tringle \(\Delta A B C\), prove that \(\frac{a^{2} \sin (B-C)}{\sin A}+\frac{b^{2} \sin (C-A)}{\sin B}+\frac{c^{2} \sin (C-A)}{\sin C}=0\)

In any triangle \(A B C\), prove that, \(r^{2}+r_{1}^{2}+r_{2}^{2}+r_{3}^{2}=16 R^{2}-\left(a^{2}+b^{2}+c^{2}\right)\)

In a triangle \(A B C, 2 a^{2}+4 b^{2}+c^{2}=4 a b+2 a c\), then the numerical value of \(\cos B\) is (a) 0 (b) \(\frac{3}{8}\) (c) \(\frac{5}{8}\) (d) \(\frac{7}{8}\)

In any triangle \(A B C\), prove that, \(\left(\left(r+r_{1}\right) \tan \left(\frac{B-C}{2}\right)\right)+\left(\left(r+r_{2}\right) \tan \left(\frac{C-A}{2}\right)\right)\) \(+\left(\left(r+r_{3}\right) \tan \left(\frac{C-A}{2}\right)\right)=0\)

If \(a, b, c\) be the sides of \(\Delta A B C\) and if roots of the equation \(a(b-c) x^{2}+b(c-a) x+c(a-b)=0\) are equal then \(\sin ^{2}\left(\frac{A}{2}\right) \cdot \sin ^{2}\left(\frac{B}{2}\right) \cdot \sin ^{2}\left(\frac{C}{2}\right)\) are in (a) A.P. (b) G.P. (c) H.P. (d) A.G.P.

In any triangle \(A B C\), prove that, \(\frac{\tan \left(\frac{A}{2}\right)}{(a-b)(a-c)}+\frac{\tan \left(\frac{B}{2}\right)}{(b-a)(b-c)}+\frac{\tan \left(\frac{C}{2}\right)}{(c-a)(c-b)}=\frac{1}{\Delta}\)

In a triangle \(A B C,(a+b+c)(b+c-a)=k \mathrm{bc}\) if (a) \(k<0\) (b) \(k>6\) (c) \(04\)

If in a \(\triangle A B C \frac{\sin A}{\sin C}=\frac{\sin (A-B)}{\sin (B-C)}\), then prove that, \(a^{2}, b^{2}, c^{2}\) are in A.P.

\(a^{3} \cos (B-C)+b^{3} \cos (C-A)+c^{3} \cos (A-B)\) is equal (a) \(3 \overrightarrow{a b c}\) (b) \((a+b+c)\) (c) \(a b c(a+b+c)\) (d) 0

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

In triangle \(A B C\), prove that, \(\frac{\text { area of the in-circle }}{\text { area of triangle } A B C}\) \(=\frac{\pi}{\cot \left(\frac{A}{2}\right) \cdot \cot \left(\frac{B}{2}\right) \cdot \cot \left(\frac{C}{2}\right)}\)

In a tringle \(\Delta A B C\), prove that \((a-b)^{2} \cos ^{2}\left(\frac{C}{2}\right)+(a+b)^{2} \sin ^{2}\left(\frac{C}{2}\right)=c^{2}\)

If in a \(\Delta A B C, \cos A+2 \cos B+\cos C=2\), then \(a, b\) \(c\) are in (a) A.P. (b) G.P. (c) H.P. (d) None

With usual notation, if in a triangle \(A B C\), \(\frac{b+c}{11}=\frac{c+a}{12}=\frac{a+b}{13}\), then prove that, \(\frac{\cos A}{7}=\frac{\cos B}{19}=\frac{\cos C}{25}\)

If the circumference of the \(\Delta A B C\) lies on its incircle, then prove that, \(\cos A+\cos B+\cos C=\sqrt{2}\)

Let \(a, b\) and \(c\) be the sides of a \(\Delta A B C\). If \(a^{2}, b^{2} \& c^{2}\) are the roots of the equation \(x^{3}-P x^{2}+Q x-R=0\), where \(p, q\) and \(R\) are constants, then find the value of \(\frac{\cos A}{a}+\frac{\cos B}{h}+\frac{\cos C}{c}\), in terms of \(P, Q\) and \(R\).

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

Let \(O\) be the circumcenter and \(H\) be the orthocenter of \(\Delta A B C\). If \(Q\) is the mid-point of \(O H\), then show that \(A Q=\frac{R}{2} \sqrt{1+8 \cos A \cos B \cos C}\)

Let \(A_{0} A_{1} A_{2} A_{3} A_{4} A_{5}\) be a regular hexagon inscribed in a circle of unit radius. The product of length of the line segment \(A_{0} A_{1}, A_{0} A_{2}, A_{0} A_{4}\) is (a) \(\frac{3}{4}\) (b) \(3 \sqrt{3}\) (c) 3 (d) \(\frac{3 \sqrt{3}}{2}\)

In a tringle \(\Delta A B C\), prove that \(\frac{\sin B}{\sin C}=\frac{c-a \cos B}{b-a \cos C}\)

If \(I_{1}, I_{2} \& I_{3}\) are the centres of escribed circles of \(\Delta A B C\), prove that the area of \(\Delta I_{1} I_{2} I_{3}=\frac{a b c}{2 r}\).

In a \(\Delta A B C\), the value of \(\frac{a \cos A+b \cos B+c \cos C}{a+b+c}\) is (a) \(\frac{R}{r}\) (b) \(\frac{R}{2 r}\) (c) \(\frac{r}{R}\) (d) \(\frac{2 r}{R}\)

In a \(\triangle A B C\), prove that, \(2\left[a \sin ^{2}\left(\frac{C}{2}\right)+c \sin ^{2}\left(\frac{A}{2}\right)\right]=c+a-b\)

In \(\Delta A B C\), prove that, \(a^{2}(s-a)+b^{2}(s-b)+c^{2}(s-c)\) \(=4 R \Delta\left(1-4 \sin \left(\frac{A}{2}\right) \sin \left(\frac{B}{2}\right) \sin \left(\frac{C}{2}\right)\right)\)

In a \(\Delta A B C\), the sides \(a, b, c\) are the roots of the equation \(x^{3}-11 x^{2}+38 x-40=0 .\) Then \(\frac{\cos A}{a}+\frac{\cos B}{b}+\) \(\frac{\cos C}{c}\) is (a) 1 (b) \(\frac{3}{4}\) (c) \(\frac{9}{16}\) (d) None

In a tringle \(\triangle A B C\), prove that \(\frac{\cos ^{2}\left(\frac{A}{2}\right)}{a}+\frac{\cos ^{2}\left(\frac{B}{2}\right)}{b}+\frac{\cos ^{2}\left(\frac{C}{2}\right)}{c}=\frac{s^{2}}{a b c}\)

Let \(O\) be a point inside a triangle \(A B C\) such that \(\angle O A B=\angle O B C=\angle O C A=\omega\), then prove that (i) \(\cot A+\cot B+\cot C=\cot \omega\) (ii) \(\operatorname{cosec}^{2} A+\operatorname{cosec}^{2} B+\operatorname{cosec}^{2} C=\operatorname{cosec}^{2} \omega\)

The ex-radii of a \(\Delta r_{1}, r_{2}, r_{3}\) are in A.P., then the sides \(a, b, c\) are in (a) A.P. (b) G.P. (c) H.P. (d) A.G.P.

In a \(\Delta A B C\), prove that \(\left(b c \cos ^{2}\left(\frac{A}{2}\right)+c a \cos ^{2}\left(\frac{B}{2}\right)+a b \cos ^{2}\left(\frac{C}{2}\right)\right)\) \(=\frac{1}{4}(a+b+c)^{2}\)

Find the distance between the circum-center and the mid-points of the sides of a triangle.

In any \(\Delta A B C, \sum \frac{\sin ^{2} A+\sin A+1}{\sin A}\) is always greater than (a) 9 (b) 3 (c) 27 (d) 36

In a \(\Delta A B C\), prove that, \(\begin{aligned}(b-c) \cot \left(\frac{A}{2}\right)+(c-a) \cot \left(\frac{B}{2}\right) & \\ &+(a-b) \cot \left(\frac{C}{2}\right)=0 \end{aligned}\)

If the sides \(a, b, c\) of a triangle are in A.P., then find the value of \(\tan \left(\frac{A}{2}\right)+\tan \left(\frac{C}{2}\right)\) in terms of \(\cot \left(\frac{B}{2}\right)\)

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 a \(\Delta A B C, a=2 b\) and \(|a-b|=\frac{\pi}{3}\), then \(\angle C\) is (a) \(\frac{\pi}{4}\) (b) \(\frac{\pi}{3}\) (c) \(\frac{\pi}{6}\) (d) None

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

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.

Prove that the distance between the circum-center and the orthocenter of a triangle is \(O H=\) \(R \sqrt{1-8 \cos A \cdot \cos B \cdot \cos C}\)