With usual rotation, in any ΔABC which of the following is/are true...?
a) Ex radius opposite to B, altitude through A, Ex radius opposite to C are in H.P.
b) If the sines of angles A,B,C are in A.P. then the altitudes of the triangle respectively are in H.P.
c) a2+b2+c2 = (r1-r)(r2+r3) + (r2-r)(r3+r1) + (r3-r)(r1+r2)
d) Origin lie inside equilateral triangle ABC. Distances of origin from sides are 4√3, 5√3, 7√3 then length of the side of the triangle is 32
[This is not my doubt... So dun juzz give hints, give proper solution... n i request nishant bhaiya n other mathematics faculty to answer it only after evry1 tries]
cheers...!!!
1c) is true
a^2 =4R^{2}sin^{2}A \\ (r_1-r)(r_2+r_3)=4Rsin(A/2)cos(\frac{B+C}{2})4Rcos(A/2)sin(\frac{B+C}{2}) \\ =4R^{2}sinAsin(B+C) \\ = 4R^{2} sin^{2}A \\ \text{Similarly we can prove for other terms,so,their sum will be equal,hence its true}
1a)We are asked to prove \frac{1}{r_2}+\frac{1}{r_3}=\frac{1}{2h} [\text{this can be understood if we draw the figure]} \\ \text{Consider LHS,} \\ \frac{r_2r_3}{r_2+r_3} \\ \\ =r_2=\frac{\triangle}{s-b} \\ r_3=\frac{\triangle}{s-c} \\ \text{After solving we get} \frac{\triangle}{a} \\ Area \triangle=(1/2)ah \\ 1/2h=\triangle/a \text{Therefore they are in HP,its true}
1if a,b,c, are in H.P. then 1/a, 1/b, 1/c are in A.P. rite...!!!
so its 2/b = 1/a + 1/c... :)
1[3]
well, maine clearly is liye kahaa ki sab doubts ek he jhatke mein chale jaaye... its safer to b conventional rite...!!! [6]
