Chapter 7

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 \(C=4 B\)

Prove that the area of an ex-central triangle is \(8 R^{2} \cos \left(\frac{A}{2}\right) \cos \left(\frac{B}{2}\right) \cos \left(\frac{C}{2}\right)\).

If the angles of a triangle are in the ratio \(1: 2: 3\), then the corresponding sides are in the ratio (a) \(2: 3: 1\) (b) \(\sqrt{3}: 2: 1\) (c) \(2: \sqrt{3}: 1\) (d) \(1: \sqrt{3}: 2\)

In any triangle \(\Delta A B C\), prove that, \(R r(\sin A+\sin B+\sin C)=\Delta .\)

Prove that the circum-radius of an ex-central triangle is \(\frac{I_{2} I_{3}}{2 \sin \left(I_{1} I_{2} I_{3}\right)}=2 R\).

In a \(\Delta A B C, a \cot A+b \cot B+c \cot C\) is (a) \(r+R\) (b) \(r-R\) (c) \(2(r+R)\) (d) \(2(r-R)\)

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

Prove that the distance between the in-center and the ex-centers are $$ \begin{gathered} I I_{1}=4 R \sin \left(\frac{A}{2}\right), I I_{2}=4 R \sin \left(\frac{B}{2}\right) \\ I I_{3}=4 R \sin \left(\frac{C}{2}\right) \end{gathered} $$

If \(A, A_{1}, A_{2}, A_{3}\) are the areas of in circle and the ex circles of a triangle, then \(\frac{1}{\sqrt{A_{1}}}+\frac{1}{\sqrt{A_{2}}}+\frac{1}{\sqrt{A_{3}}}\) is (a) \(\frac{2}{\sqrt{A}}\) (b) \(\frac{1}{\sqrt{A}}\) (c) \(\frac{1}{2 \sqrt{A}}\) (d) \(\frac{3}{\sqrt{A}}\)

In any triangle \(\Delta A B C\), prove that, \(\frac{1}{b c}+\frac{1}{c a}+\frac{1}{a b}=\frac{1}{2 R r}\)

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

In any triangle \(\Delta A B C\), prove that, \(\cos ^{2}\left(\frac{A}{2}\right)+\cos ^{2}\left(\frac{B}{2}\right)+\cos ^{2}\left(\frac{C}{2}\right)=2+\frac{r}{2 R}\)

In any triangle \(A B C\), prove that, \(a^{3} \cos (B-C)+b^{3} \cos (C-A)\) $$ +c^{3} \cos (A-B)=3 a b c $$

In any triangle \(\triangle A B C\), prove that, \(a \cot A+b \cot B+c \cot C=2(R+r)\)

The sides of a triangle are in A.P. and the greatest and least angles are \(\theta\) and \(\phi\), respectively, then prove that \(4(1-\cos \theta)(1-\cos \varphi)=\cos \theta+\cos \varphi\)

In a \(\triangle A B C, \tan A \tan B \tan C=9 .\) For such triangles, if \(\tan ^{2} A+\tan ^{2} B+\tan ^{2} C=\lambda, 7\) then (a) 9. \(\sqrt[3]{3}<\lambda<27\) (b) \(\lambda \leq 27\) (c) \(\lambda<9 . \sqrt[3]{3}\) (d) \(\lambda>27\)

In a \(\Delta A B C, a^{2} \cos ^{2} A=b^{2}+c^{2}\), then (a) \(A<\frac{\pi}{4}\) (b) \(\frac{\pi}{4}\frac{\pi}{2}\) (d) \(A=\frac{\pi}{2}\)

In any triangle \(\Delta A B C\), prove that, prove that, \(\left(\frac{1}{r}+\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}\right)^{2}=\frac{4}{r}\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}\right)\)

Prove that the triangle having sides \(3 x+4 y, 4 x+3 y\) and \(5 x+5 y\) units, respectively, where \(x, y>0\), is obtuse angled.

If \(A, B, C\) are angles of a triangle such that the angle \(A\) is obtuse, then \(\tan B \tan C<\) (a) 0 (b) 1 (c) 2 (d) 3

If \(\Delta\) be the area and ' \(s\) ' be the semi perimeter of a triangle, then prove that, \(\Delta \leq \frac{s^{2}}{3 \sqrt{3}}\)

Prove that the radii of the three escribed circles of a triangle are the roots of \(x^{3}-(r+4 R) x^{2}+s^{2} x-r s^{2}=0\)

Let \(A B C\) be a triangle having altitudes \(h_{1}, h_{2}\) \& \(h_{3}\) from the vertices \(A, B, C\), respectively, and \(r\) be the in-radius, prove that, \(\frac{h_{1}+r}{h_{1}-r}+\frac{h_{2}+r}{h_{2}-r}+\frac{h_{3}+r}{h_{3}-r} \geq 6\).

If \(\alpha, \beta \& \gamma\) are the respective altitudes of a triangle \(A B C\), prove that \(\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}+\frac{1}{\gamma^{2}}=\frac{\cot A+\cot B+\cot C}{\Delta}\)

Two circles of radii \(a\) and \(b\) cut each other at an angle \(\theta\). Prove that the length of the common chord is \(\frac{2 a b \sin \theta}{\sqrt{a^{2}+b^{2}+2 a b \cos \theta}}\)

If \(\alpha, \beta \& \gamma\) are the respective altitudes of a triangle \(A B C\), prove that \(\frac{1}{\alpha}+\frac{1}{\beta}-\frac{1}{\gamma}=\left(\frac{2 a b}{(a+b+c) \Delta} \times \cos ^{2}\left(\frac{C}{2}\right)\right)\)

If \(\alpha, \beta, \gamma\) are the distances of the vertices of a triangle from the corresponding points of contact with the in-circle, prove that \(r^{2}=\frac{\alpha \beta \gamma}{\alpha+\beta+\gamma}\)

If \(c^{2}=a^{2}+b^{2}, 2 s=a+b+c\), then \(4 s(s-a)(s-b)(s-c)\) is (a) \(s^{4}\) (b) \(b^{2} c^{2}\) (c) \(c^{2} a^{2}\) (d) \(a^{2} b^{2}\)

Tangents are drawn to the in-circle of a triangle \(A B C\) which are parallel to its sides. If \(x, y, z\) be the lengths of the tangents and \(a, b, c\) be the sides of a triangle, hen prove that, \(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1\)

The sides of a triangle are \(x^{2}+x+1,2 x+1\) and \(x^{2}-1\), prove that the greatest angle is \(120^{\circ} .\)

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the side of the triangle.

if \(x, y, z\) be the lengths of the perpendiculars from the circumcentre on the sides \(B C, C A, A B\) of a triangle \(A B C\), prove that, \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=\frac{a b c}{4 x y z}\)

\(A B C\) is a triangle. Its area is 12 sq. \(\mathrm{cm}\) and base is 6 \(\mathrm{cm}\). The difference of the angles is \(60^{\circ}\). Prove that the angle \(A\) opposite to the base is given by \(8 \sin A-6\) \(\cos A=3\)

If \(p_{1}, p_{2}, p_{3}\) are the altitudes of the triangle \(A B C\) from the vertices \(a, b\) and \(c\), respectively, prove that \(\frac{\cos A}{p_{1}}+\frac{\cos B}{p_{2}}+\frac{\cos C}{p_{3}}=\frac{1}{R}\)

In any triangle \(A B C\), if \(\cos \theta=\frac{a}{b+c}\), \(\cos \varphi=\frac{b}{a+c}, \cos \psi=\frac{c}{a+b}\), where \(\theta, \varphi \& \psi\) lie between 0 and \(\pi\), prove that, \(\tan ^{2}\left(\frac{\theta}{2}\right)+\tan ^{2}\left(\frac{\varphi}{2}\right)+\tan ^{2}\left(\frac{\psi}{2}\right)=1\)

The product of the sines of the angles of \(a\) triangle is \(p\) and the product of their cosines is \(q\). Prove that the tangents of the angles are the roots of \(q x^{3}-p x^{2}+(1+q) x-p=0 .\)

The sides of a triangle are \(A B C\) are in A.P. If the angles a and \(c\) are the greatest and the smallest angles respectively, then prove that, \(4(1-\cos A)(1-\cos C)=\cos A+\cos C\)

In a triangle \(A B C\), if \(\cos A \cdot \cos B+\sin A \sin B \sin C\) \(=1\), prove that the sides are in the ratio \(1: 1: \sqrt{2}\).

In triangle \(A B C\), if \(a, b, c\) are in H.P., prove that \(\sin ^{2}\left(\frac{A}{2}\right), \sin ^{2}\left(\frac{B}{2}\right), \sin ^{2}\left(\frac{C}{2}\right)\) are also in H.P.

The base of a triangle is divided into three equal parts If \(t_{1}, t_{2} \& t_{3}\) be the tangents of the angles subtended by these parts at the opposite vertices, prove that \(\left(\frac{1}{t_{1}}+\frac{1}{t_{2}}\right)\left(\frac{1}{t_{2}}+\frac{1}{t_{3}}\right)=4\left(1+\frac{1}{t_{2}^{2}}\right)\)

In a triangle \(A B C\), if the sides \(a, b, c\) be in A.P., prove that \(\cos A \cdot \cot \left(\frac{A}{2}\right), \cos B \cdot \cot \left(\frac{B}{2}\right), \cos C \cdot \cot \left(\frac{C}{2}\right)\) are also in A.P.

The sides of a triangle are in A.P. and its area is \(3 / 5\) th of an equilateral triangle of the same perimeter. Prove that the sides are in the ratio \(3: 5: 7\).

The ex-radii \(r_{1}, r_{2}, r_{3}\) of a triangle \(A B C\) are in H.P., prove that the sides \(a, b, c\) are in A.P.

The diameter of the circum-circle of a triangle with sides \(5 \mathrm{~cm}, 6 \mathrm{~cm} ., 7 \mathrm{~cm}\) is (a) \(\frac{3 \sqrt{6}}{2} \mathrm{~cm}\). (b) \(2 \sqrt{6} \mathrm{~cm}\) (c) \(\frac{35}{48} \mathrm{~cm}\) (d) \(\frac{35}{2 \sqrt{6}}\)

If \(A, B, C\) are the angles of a triangle, then prove that \(\cos A+\cos B+\cos C=1+\frac{r}{R}\), where \(r=\) in-radius and \(R=\) circum - radius.

In a triangle \(A B C\), the measures of the angles \(A, B, C\) are \(3 \alpha, 3 \beta\) and \(3 \gamma\), respectively. \(P, Q\), and \(R\) are the points within the triangle such that \(\angle B A R=\angle R A Q=\angle Q A C=\alpha\), \(\angle C B P=\angle P B R=\angle R B A=\beta\) and \(\angle A C Q=\angle Q C P=\angle P C B=\gamma\), then prove that \(A R=8 R \sin \beta \sin \gamma \cos \left(30^{\circ}-\gamma\right)\)

In a \(\Delta A B C\), the sides are in the ratio \(4: 5: 6\). The ratio of the circum-radius and the in-radius is (a) \(8: 7\) (b) \(3: 2\) (c) \(7: 3\) (d) \(16: 7\)

In triangle \(A B C\), if \(8 R^{2}=a^{2}+b^{2}+c^{2}\), prove that the triangle is right angled.

If in a triangle \(A B C\), the median \(A D\) and the perpendicular \(A E\) from the vertex \(A\) to the side \(B C\) divides the angle \(A\) into three equal parts, show that \(\cos \left(\frac{A}{3}\right) \cdot \sin ^{2}\left(\frac{A}{3}\right)=\frac{3 a^{2}}{32 b c}\).

If in a triangle, \(R\) and \(r\) are the circum radius and in radius, respectively, then the H.M. of the ex-radii of the triangle is (a) \(3 r\) (b) \(2 R\) (c) \(R+r\) (d) None