`T_1` is an isosceles triangle in circle C. Let `T_2` be another ísosceles triangle inscribed in C whose base is one of the equal sides of `T_1` and which overlaps the interior of `T_1`. Similarly, create isosceles triangle `T_3` from `T_2; T_4 and T_5`, and so on. Prove that the triangle `T_n`, approaches an equilateral triangle as `n ->oo`,

Show that the relation R defined in the set A of all triangles as : R = [(T_1, T_2): T_1 and T_2 are similar triangles} is an equivalence relation .

IF T_(n) dentoes the number of triangle formed with n points in plane no three of which are collinear and If T_(n+1) - T_(n)= 36 then n=

In triangle A B C , if sinAcosB=1/4 and 3t a n A=t a n B ,t h e ncot^2A is equal to

An equilateral triangle, a square and a circle have equal perimeters. If T denotes the area of the triangle, S , the area of the square and C , the area of the circle, then S

ABCD is a square with side AB = 2. A point P moves such that its distance from A equals its distance from the line BD. The locus of P meets the line AC at T_1 and the line through A parallel to BD at T_2 and T_3 . The area of the triangle T_1 T_2 T_3 is :

ABCD is a square with side AB = 2. A point P moves such that its distance from A equals its distance from the line BD. The locus of P meets the line AC at T_1 and the line through A parallel to BD at T_2 and T_3 . The area of the triangle T_1 T_2 T_3 is :

In Figure (P S)/(S Q)=(P T)/(T R) and /_P S T=/_P R Q . Prove that PQR is an isosceles triangle.

Let T bt the set of all triangle in the coordinate plane. Show that the relations in T defined as R_1 = {(T_1,T_2) : T_1, T_2 in T} T_1 is similar to T_2 and R_2 = {(T_1, T_2) : T_1, T_2 in T} , T_1 is congruent to T_2 , are equivalence relations.

Isosceles triangle T_1 has a base of 12 meters and a height of 20 meters . The vertices of a second triangle T_2 are the midpoints of the sides of T_1 . The vertices of a third triangle , T_3 , are the midpoints of the sides of T_2 . Assume the process continues indefinitely , with the vertices of T_(k+1) being the midpoints of the sides of T_k for every positive integer k. What is the sum of the areas, in square meters, of T_1,T_2,T_3 , ..... ?