Chapter 13

PROVING IDENTITIES RELATED TO EX-RADII $$ \frac{(b-c)}{r_{1}}+\frac{(c-a)}{r_{2}}+\frac{(a-b)}{r_{3}}=0 $$

PROVING IDENTITIES RELATED TO EX-RADII $$ \frac{\left(r_{2}+r_{3}\right)}{(1+\cos A)}=\frac{\left(r_{3}+r_{1}\right)}{1+\cos B}=\frac{\left(r_{1}+r_{2}\right)}{1+\cos C} $$

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

Prove that i. \(\quad A I_{1}=r_{1} \operatorname{cosec} \frac{A}{2}\) ii. \(I I_{1}=a \sec \frac{A}{2}\) iii. \(I_{2} I_{3}=a \operatorname{cosec} \frac{A}{2}=4 R \cos \frac{A}{2}\) iv. \(I I_{1} \cdot I I_{2} \cdot I I_{3}=16 R^{2} r\) v. \(\quad I_{2} I_{3}{\underline{\phantom{xx}}}^{2}=4 R\left(r_{2}+r_{3}\right)\) vi. \(\quad \angle I_{3} I_{1} I_{2}=\frac{B+C}{2}=\frac{\pi}{2}-\frac{A}{2}\) vii. \(I I_{1}{\underline{\phantom{xx}}}^{2}+I_{2} I_{3}{\underline{\phantom{xx}}}^{2}=I I_{2}{\underline{\phantom{xx}}}^{2}+I_{3} I_{1}{\underline{\phantom{xx}}}^{2}=I I_{3}{\underline{\phantom{xx}}}^{2}+I_{1} I_{2}{\underline{\phantom{xx}}}^{2}\) viii. Area of \(\Delta I_{1} I_{2} I_{3}=8 R^{2} \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}=\frac{a b c}{2 r}=2 R s\) ix. \(\quad \frac{I I_{1} \cdot I_{2} I_{3}}{\sin A}=\frac{I I_{2} \cdot I_{3} I_{1}}{\sin B}=\frac{I I_{3} \cdot I_{1} I_{2}}{\sin C}\)

If \(x, y, z\) be the lengths of the tangents from the centers of ex-circles to the circumcircle, prove that \(\frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=\frac{2 s}{a b c}\)

If \(\Delta_{0}\) be the area of the triangle formed by joining the points of contact of the incircle with the sides of the given triangle and \(\Delta_{1}, \Delta_{2}\) and \(\Delta_{3}\) the corresponding areas for the ex-circles, prove that \(\Delta_{1}+\Delta_{2}+\Delta_{3}-\Delta_{0}=2 \Delta\)

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

Three equal circles touch one another. Find the radius of the circle which touches all three. \\{Ans. \(\left.\left(\frac{2}{\sqrt{3}} \pm 1\right) r\right\\}\)

Three circles, whose radii are \(a, b\) and \(c\), touch one another externally and the tangents at their points of contact meet in a point. Prove that the distance of this point from either of their points of contact is \(\left(\frac{a b c}{a+b+c}\right)^{\frac{1}{2}}\)

Three circles touch one another externally. The tangents at their points of contact meet at a point whose distance from any point of contact is 4 . Find the ratio of the product of the radii to the sum of the radii of circles.

In an acute angled triangle, prove that \(\tan A+\tan B+\tan C \geq 3 \sqrt{3}\). If \(\tan A+\tan B+\tan C=3 \sqrt{3}\), prove that the triangle is equilateral.

Prove that \(\cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2} \geq 3 \sqrt{3}\)

Prove that \(\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \leq \frac{1}{8}\).

Prove that the area of any quadrilateral is one-half the product of the two diagonals and the sine of the angle between them.

If \(a, b, c\) and \(d\) be the sides and \(x\) and \(y\) the diagonals of a quadrilateral, prove that it's area is \(\frac{1}{4} \sqrt{4 x^{2} y^{2}-\left(b^{2}+d^{2}-a^{2}-c^{2}\right)^{2}}\)

If \(A B C D\) be a cyclic quadrilateral, prove that \(\tan \frac{B}{2}=\sqrt{\frac{(s-a)(s-b)}{(s-c)(s-d)}}\).

The sides of a quadrilateral with an inscribed circle are \(7,10,5\) and \(2 \mathrm{~cm}\). and the sum of a pair of opposite angles is \(120^{\circ}\). Find area and radius of inscribed circle.

A quadrilateral \(A B C D\) is circumscribed about a circle, prove that \(a \sin \frac{A}{2} \sin \frac{B}{2}=c \sin \frac{C}{2} \sin \frac{D}{2}\).

Let \(A B C\) be a triangle with incentre \(I\) and inradius \(r\). Let \(D, E, F\) be the feet of the perpendiculars from \(I\) to the sides \(B C, C A\) and \(A B\) respectively. If \(r_{1}, r_{2}\) and \(r_{3}\) are the radii of circles inscribed in the quadrilateral \(A F I E, B D I F\) and \(C E I D\) respectively, prove that \(\frac{r_{1}}{r-r_{1}}+\frac{r_{2}}{r-r_{2}}+\frac{r_{3}}{r-r_{3}}=\frac{r_{1} r_{2} r_{3}}{\left(r-r_{1}\right)\left(r-r_{2}\right)\left(r-r_{3}\right)} .\)

Find the length of the side of a regular polygon of 12 sides which is circumscribed to a circle of unit radius.

Find the area of a pentagon, a hexagon, an octagon and a decagon, each being a regular figure of side 1 meter.

Find the difference between the areas of a regular octagon and a regular hexagon if the perimeter of each is \(24 \mathrm{~cm}\).

Compare the areas and perimeters of octagons which are respectively inscribed in and circumscribed to a given circle.

If an equilateral triangle and a regular hexagon have the same perimeter, prove that their areas are as \(2: 3\).

Given that the area of a polygon of \(n\) sides circumscribed about a circle is to the area of the circumscribed polygon of \(2 n\) sides as \(3: 2\), find \(n\).

The area of a regular polygon of \(n\) sides inscribed in a circle is to that of the same number of sides circumscribing the same circle as \(3: 4\). Find the value of \(n\).

The interior angles of a polygon are in A.P., the least angle is \(120^{\circ}\) and the common difference is \(5^{\circ} .\) Find the number of sides.

There are two regular polygons, the number of sides in one being double the number in the other, and an angle of one polygon is to an angle of the other as \(9: 8\), find the number of sides of each polygon.

Show that there are eleven pairs of regular polygons such that the number of degrees in the angle of one is to the number in the angle of the other as \(10: 9\). Find the number of sides in each.

Prove that the area of the circle and the area of a regular polygon of \(n\) sides and of perimeter equal to that of the circle are in the ratio of \(\tan \frac{\pi}{n}: \frac{\pi}{n}\).

Prove that the sum of the radii of the circles, which are respectively in and circumscribed about a regular polygon of \(n\) sides, is \(\frac{a}{2} \cot \frac{\pi}{2 n}\), where \(a\) is a side of the polygon.

Of two regular polygons of \(n\) sides, one circumscribed and the other is inscribed in a given circle, prove that the perimeters of the circumscribing polygon, the circle and the inscribed polygon are in the ratio \(\sec \frac{\pi}{n}: \frac{\pi}{n} \operatorname{cosec} \frac{\pi}{n}: 1\) and that the areas of the polygons are in the ratio \(\cos ^{2} \frac{\pi}{n}: 1\).

Prove that the area of a regular polygon of \(2 n\) sides inscribed in a circle is a mean proportional between the areas of the regular inscribed and circumscribed polygons of \(n\) sides.

A pyramid stands on a regular hexagon as base. The perpendicular from the vertex of the pyramid on the base passes through the center of the hexagon and it's length is equal to that of a side of the base. Find the tangent of the angle between the base and any face of the pyramid and also of half the angle between any two side faces.

A regular pyramid has for it's base a polygon of \(n\) sides and each slant face consists of an isosceles triangle of vertical angle \(2 \alpha\). If the slant faces are each inclined at an angle \(\beta\) to the base and at an angle \(2 \gamma\) to one another, show that \(\cos \beta=\tan \alpha \cot \frac{\pi}{n}\) and \(\cos \gamma=\sec \alpha \cos \frac{\pi}{n}\).

If \(A, B, C\) are the angles of a triangle, then show that the system of equations \(-x+y \cos C+z \cos B=0\) \(x \cos C-y+z \cos A=0\) \(x \cos B+y \cos A-z=0 .\) has non-zero solution.

If \(A, B, C\) are the angles of a triangle show that system of equations \(x \sin 2 A+y \sin C+z \sin B=0\) \(x \sin C+y \sin 2 B+z \sin A=0\) \(x \sin B+y \sin A+z \sin 2 C=0\) possesses non-trivial solution.