Home
Class 12
MATHS
Perpendiculars are drawn from the angles...

Perpendiculars are drawn from the anglesA, B,C,of an acute angles `Delta` on the opposite sodes and products to meet the circumscribing circle. If these produced parts be `propto,beta,gamma`respectively, show that
`(a)/prop+(b)/beta+(c)/gamma=2(tanA+tanB+tanC`

Text Solution

Verified by Experts

Let AD be the perpendicular from A on BC. When AD is produced, it meets the circumscribing circle at E. From question `DE = alpha`.

Since angle in the same segment are equal, we have
`angle AEB = angleACB = C`
and `angle AEC = angle ABC = B`
From the right-angled triangle BDE
`tan C = (BD)/(DE)`...(i)
From the right-angled triangle CDE,
`tan B = (CD)/(DE)`...(ii)
Adding Eqs. (i) and (ii), we get
`tan B + tan C = (BD + CD)/(DE) = (BC)/(DE) = (a)/(alpha)`....(iii)
Similarly, `tan C + tan A = (b)/(beta)`...(iv)
and `tan A + tan B = (c)/(gamma)`...(v)
Eqs. (iii), (iv), and (v), we get
`(a)/(alpha) + (b)/(beta) + (c)/(gamma) = 2 (tan A + tan B + tan C)`
Promotional Banner

Topper's Solved these Questions

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Illustration|86 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Concept application exercise 5.1|12 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise ARCHIVES (NUMERICAL VALUE TYPE )|8 Videos

  • RELATIONS AND FUNCTIONS

    CENGAGE PUBLICATION|Exercise All Questions|1119 Videos

Similar Questions

Explore conceptually related problems

In A B C , the bisector of the angle A meets the side BC at D andthe circumscribed circle at E. Prove that D E=(a^2secA/2)/(2(b+c))

The perpendicular from the vertices A, B and C of a triangle ABC on the opposite sides meet at O. If OA=x, OB=y and OC=z, prove that, a/x + b/y + c/z = (abc)/(xyz) .

The corodinates of the three vertices of a triangle are (a, a tan alpha) , (b ,b tan beta) and (c, c tangamma) . If the circumcentre and orthocentre of the triangle are at (0,0) and (h,k) respectively, prove that (h)/(k)=(cos alpha+ cos beta+ cos gamma)/( sin alpha+ sin beta + sin gamma) .

If x,y,z are the perpendiculars from the vertices of a triangle ABC on the opposite sides a,b,c respectively, then show that (bx)/c+(cy)/a+(az)/b=(a^2+b^2+c^2)/(2R)

Two medians drawn from the acute angles of a right angled triangle intersect at an angle pi/6. If the length of the hypotenuse of the triangle is 3 units, then the area of the triangle (in sq. units) is (a) sqrt3 (b) 3 (c) sqrt2 (d) 9

Perpendicular drawn from the vertices A,B,C upon opposite sides of triangle ABC passes through a fixed point O.if bar(OA)=x.bar(OB)=y and bar(OC)=z then prove that a/x+b/y+c/z=(abc)/(xyz)

Consider an acute angled Delta ABC. Let AD, BE and CF be the altitudes drawn from the vertice to the opposite sides. Prove that : (EF)/(a)+ (FD)/(b)+ (DE)/(c)= (R+r)/(R ).

The diameters of the circumcirle of triangle ABC drawn from A,B and C meet BC, CA and AB, respectively, in L,M and N. Prove that (1)/(AL) + (1)/(BM) + (1)/(CN) = (2)/(R)

Let A B C be a triangle right-angled at Aa n dS be its circumcircle. Let S_1 be the circle touching the lines A B and A C and the circle S internally. Further, let S_2 be the circle touching the lines A B and A C produced and the circle S externally. If r_1 and r_2 are the radii of the circles S_1 and S_2 , respectively, show that r_1r_2=4 area ( A B C)dot

The regular three dimensional arrangement of points in a crystal is known as crystal lattice and the smallest repeating pattern in the lattice is called unit cell. The unit cells are characterised by the edge lengths a, b, c and the angles between them alpha, beta and gamma respectively. Based on this, there are seven crystal systems. In a cubic unit cell: a=b=c and alpha = beta=gamma=90^(@) The number of points in simple, body centred and face centred cubic cells are 1,2 and 4 respectively In both the hcp and ccp of spheres, the number of tetrahedral voids per sphere is two while the octahedral voids is one. The C.N of cation occuppying an octahedral vois is: