Properties of triangle of altitudes trigonometry problems

1) If p1, p2, p3 are the altitudes of triangle ABC then show that 1/p1^2+1/p2^2+1/p3^2= cot A+cotB+cotC/Delta
2) If p1, p2, p3 are the altitudes drawn from vertices A, B, C to the opposite sides of triangle respectively then show that (i) 1/p1+1/p2+1/p3=1/r (ii) p1p2p3=(ABC) ^2/8R^3
3) Prove that a^3cos(B-C) +b^3cos(C-A) +c^3cos(A-B) =3abc
4)Prove that 4(r1r2+r2r3+r3r1)=(a+b+c)^2
5)If cotA/2,cotB/2,cotC/2 are in A.P. , then prove that a,b,c are in A.P.
#maths #maths1a
#trigonometryproblems #trigonometry