Angle `A ,\ B ,\ C` of a triangle `A B C` are equal to each other. Prove that ` A B C` is equilateral.

A D ,\ B E\ a n d\ C F , the altitudes of A B C are equal. Prove that A B C is an equilateral triangle.

A D ,\ B E\ a n d\ C F , the altitudes of A B C are equal. Prove that A B C is an equilateral triangle.

In a triangle A B C ,\ \ A D_|_B C and A D^2=B DxxC Ddot Prove that A B C is a right triangle.

In Figure, D ,E and F are, respectively the mid-points of sides B C ,C A and A B of an equilateral triangle A B C . Prove that D E F is also an equilateral triangle.

In Figure, D ,E and F are, respectively the mid-points of sides B C ,C A and A B of an equilateral triangle A B C . Prove that D E F is also an equilateral triangle.

If A ,\ B ,\ C are the angles of a triangle, prove that : cos A+cos B+cos C=1+r/Rdot

Fill in the blanks in the following so that each of the following statements is true. (i) Sides opposite to equal angles of a triangle are ....... (ii) Angle opposite to equal sides of a triangle are ....... (iii) In an equilateral triangle all angles are ....... (iv) In a A B C if /_A=/_C , then A B=\ ..... (v) If altitudes C E\ a n d\ B F of a triangle A B C are equal, then A B= ........ (vi) In an isosceles triangle A B C with A B=A C , if B D\ a n d\ C E are its altitudes, then B D is .......... C E . (vii) In right triangles A B C\ a n d\ D E F , if hypotenuse A B=E F and side A C=D E , then A B C\ ~= ....

Diagonals A C and B D of a trapezium A B C D with A B || C D intersect each other at O . Prove that ar ( triangle A O D)=a r( triangle B O C) .

A triangle A B C is right angled at A. AL is drawn perpendicular to B C . Prove that /_B A L=/_A C Bdot

D and E are points on equal sides A B and A C of an isosceles triangle A B C such that A D=A E . Prove that B ,C ,D ,E are concylic.