91Ó°ÊÓ

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 \(2 \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}: \cos \frac{A-B}{2} \cos \frac{B-C}{2} \cos \frac{C-A}{2}\)

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

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

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