Chapter 13

If \(D\) is the mid-point of \(B C\), then prove that \(\sin \angle C A D=\frac{a \sin C}{\sqrt{2 b^{2}+2 c^{2}-a^{2}}}\), \(\sin \angle B A D=\frac{a \sin B}{\sqrt{2 b^{2}+2 c^{2}-a^{2}}}, \sin \angle A D B=\frac{2 b \sin C}{\sqrt{2 b^{2}+2 c^{2}-a^{2}}}\) and \(\cot \angle A D B=\frac{\left(b^{2}-c^{2}\right)}{4 \Delta} .\)

In a triangle \(A B C\), the median to the side \(B C\) is of length \(\frac{1}{\sqrt{11-6 \sqrt{3}}}\) and it divides angle \(A\) into angles of \(30^{\circ}\) and \(45^{\circ}\). Prove that side \(B C\) is of length 2 units.

If in a triangle the median through \(A\) is perpendicular to the side \(A B\), prove that \(\tan A+2 \tan B=0\).

If \(D\) is the mid-point of \(B C\) and \(A D\) is perpendicular to \(A C\), then prove that \(\cos A \cos C=\frac{2\left(c^{2}-a^{2}\right)}{3 a c}\).

Prove that the median through \(A\) divides it into angles whose cotangents are \(2 \cot A+\cot C\) and \(2 \cot A+\cot B\) and makes with the base an angle whose cotangent is \(\frac{1}{2}(\cot C \sim \cot B)\).

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.

The perpendicular \(A D\) to the base of a triangle \(A B C\) divides it into segments such that \(B D, C D\) and \(A D\) are in the ratio of 2,3 and \(6 ;\) prove that the vertical angle of the triangle is \(45^{\circ}\).

If \(\sin A, \sin B, \sin C\) are in A.P., then prove that the altitudes are in H.P.

If \(A D\) is the altitude from \(A, b>c, C=23^{\circ}\) and \(A D=\frac{a b c}{b^{2}-c^{2}}\), find \(B\).

Prove that the perpendicular from \(A\) divides \(B C\) into portions which are proportional to the cotangents of the adjacent angles and that it divides the angle \(A\) into portions whose cosines are inversely proportional to the adjacent sides.

Prove that the distance between the middle point of \(B C\) and the foot of the perpendicular from \(A\) is \(\frac{b^{2} \sim c^{2}}{2 a}\)

If \(p, q, r\) are the altitudes of a triangle \(A B C\), prove that \(\frac{1}{p^{2}}+\frac{1}{q^{2}}+\frac{1}{r^{2}}=\frac{\cot A+\cot B+\cot C}{\Delta}\).

If \(p_{1}, p_{2}, p_{3}\) are altitudes of a triangle \(A B C\) from the vertices \(A, B, C\) and \(\Delta\) the area of the triangle, then prove that \(p_{1}^{-1}+p_{2}^{-1}-p_{3}^{-1}=\frac{s-c}{\Delta}\).

If \(p, q, r\) are the altitudes of a triangle from the vertices \(A, B, C\) respectively, prove that \(\frac{1}{p}+\frac{1}{q}-\frac{1}{r}=\frac{a b}{s \Delta} \cos ^{2} \frac{C}{2}\)

In a triangle of base \(a\), the ratio of the other sides is \(r(r<1)\). Show that the altitude of the triangle is less than or equal to \(\frac{a r}{1-r^{2}}\).

In a right angled triangle \(A B C\), the bisector of the right angle \(C\) divides \(A B\) into segments \(p\) and \(q\) and if \(\tan \frac{A-B}{2}=t\), then show that \(p: q=(1-t):(1+t)\).

If the bisectors of the angles of a triangle \(A B C\) meet the opposite sides in \(A^{\prime}, B^{\prime}\) and \(C^{\prime}\), prove that the ratio of the areas of the triangles \(A^{\prime} B^{\prime} C^{\prime}\) and \(A B C\) is \(2 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}: \cos \frac{A-B}{2} \cos \frac{B-C}{2} \cos \frac{C-A}{2}\)

\(\Delta=2 R^{2} \sin A \sin B \sin C\)

\(4 R \sin A \sin B \sin C=a \cos A+b \cos B+c \cos C\)

\(\sin A+\sin B+\sin C=\frac{s}{R}\)

Find the circumradius of the equilateral triangle of side \(2 \sqrt{3} \mathrm{~cm}\).

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

If \(x, y, z\) are respectively the perpendiculars from the vertices \(A, B, C\) to the opposite sides, prove that i. \(x y z=\frac{a^{2} b^{2} c^{2}}{8 R^{3}}\) ii. \(\frac{\cos A}{x}+\frac{\cos B}{y}+\frac{\cos C}{z}=\frac{1}{R}\) iii. \(\frac{b x}{c}+\frac{c y}{a}+\frac{a z}{b}=\frac{a^{2}+b^{2}+c^{2}}{2 R}\)

If \(x, y, z\) are respectively the perpendiculars from the circumcentre to the sides, prove that \(\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=\frac{a b c}{4 x y z}\)

Prove that the radii of the circles circumscribing the triangles \(B P C, C P A, A P B\) and \(A B C\) are all equal.

\(D, E\) and \(F\) are the middle points of the sides of the triangle \(A B C\); prove that the centroid of the triangle \(D E F\) is the same as that of \(A B C\) and that it's orthocentre is the circumcentre of \(A B C .\)

At the points \(A, B, C\), tangents are drawn to the circumcircle. These tangents enclose a triangle \(P Q R\). Prove that its angles and sides are respectively \(180^{\circ}-2 A, 180^{\circ}-2 B, 180^{\circ}-2 C\) and \(\frac{a}{2 \cos B \cos C}\), \(\frac{b}{2 \cos C \cos A}, \frac{c}{2 \cos A \cos B}\)

The legs of a tripod are each \(10 \mathrm{~cm}\). in length and their points of contact with a horizontal table on which the tripod stands form a triangle whose sides are 7,8 and \(9 \mathrm{~cm}\). in length. Find the inclination of the legs to the horizontal and the height of the apex.

\(a \cot A+b \cot B+c \cot C=2(R+r)\)

\((b+c) \tan \frac{A}{2}+(c+a) \tan \frac{B}{2}+(a+b) \tan \frac{C}{2}=4(R+r)\)

\(\cos ^{2} \frac{A}{2}+\cos ^{2} \frac{B}{2}+\cos ^{2} \frac{C}{2}=2+\frac{r}{2 R}\)

Find the in-radius of the triangle having sides \(13,14,15\).

\(D E F\) 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. \(\frac{r \Delta}{2 R}\).

\(A D, B E\) and \(C F\) are the perpendiculars from the angular points of a triangle \(A B C\) upon the opposite sides. Prove that the diameters of the circumcircles of the triangles \(A E F, B D F\) and \(C D E\) are respectively \(a \cot A\), \(b \cot B\) and \(c \cot C\), and that the perimeters of the triangles \(D E F\) and \(A B C\) are in the ratio \(r: R\).

If \(x, y, z\) are respectively the perpendiculars from the vertices \(A, B, C\) to the opposite sides, prove that \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{r}\)

If \(x, y, z\) are respectively the distance of the vertices from orthocentre, prove that \(x+y+z=2(R+r)\).

If \(x, y, z\) are the distances of the vertices of a triangle from the corresponding points of contact with the incircle, prove that \(\frac{x y z}{(x+y+z)}=r^{2}\).

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 \(A D=E F\).

Let \(A B C\) be a triangle having \(O\) and \(I\) as its circumcentre and incentre respectively. If \(R\) and \(r\) are the circumradius and the inradius respectively, then prove that \((I O)^{2}=R^{2}-2 R r\). Further show that the triangle \(B I O\) is right-angled triangle if and only if \(b\) is the arithmetic mean of \(a\) and \(c\).

Prove that the area of the incircle is to the area of the triangle itself is \(\pi: \cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2}\).

\(C\) is right-angled and a perpendicular \(C D\) is drawn to \(A B\). The radii of the circles inscribed into the triangles \(A C D\) and \(B C D\) are equal to \(x\) and \(y\) respectively. Find the radius of the circle inscribed into the triangle \(A B C\).

If a circle be drawn touching the incircle and circumcircle of a triangle and the side \(B C\) externally, prove that it's radius is \(\frac{\Delta}{a} \tan ^{2} \frac{A}{2}\).

\(\frac{r r_{1}}{r_{2} r_{3}}=\tan ^{2} \frac{A}{2}\)

PROVING IDENTITIES RELATED TO EX-RADII $$ r r_{1} \cot \frac{A}{2}=\Delta $$

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

PROVING IDENTITIES RELATED TO EX-RADII $$ \left(r_{1}-r\right)\left(r_{2}-r\right)\left(r_{3}-r\right)=4 R r^{2} $$

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

PROVING IDENTITIES RELATED TO EX-RADII $$ \left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)\left(\frac{1}{r_{2}}+\frac{1}{r_{3}}\right)\left(\frac{1}{r_{3}}+\frac{1}{r_{1}}\right)=\frac{64 R^{3}}{a^{2} b^{2} c^{2}} $$

PROVING IDENTITIES RELATED TO EX-RADII $$ R=\frac{\left(r_{1}+r_{2}\right)\left(r_{2}+r_{3}\right)\left(r_{3}+r_{1}\right)}{4\left(r_{1} r_{2}+r_{2} r_{3}+r_{3} r_{1}\right)} $$

PROVING IDENTITIES RELATED TO EX-RADII $$ \left(r+r_{1}\right) \tan \frac{B-C}{2}+\left(r+r_{2}\right) \tan \frac{C-A}{2}+\left(r+r_{3}\right) \tan \frac{A-B}{2}=0 $$