`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.

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

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.

A D\ a n d\ B E are respectively altitudes of A B C such that A E=B D . Prove that A D=B E .

In Figure, B D and C E are two altitudes of a A B C such that B D=C Edot Prove that A B C is isosceles. Figure

In Figure, B D and C E are two altitudes of a A B C such that B D=C Edot Prove that A B C is isosceles.

In Fig. 4.101, A D and C E are two altitudes of A B C . Prove that A E F ~ C D F (ii) A B D ~ C B E

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\ ~= ....

A B C D is a parallelogram, G is the point on A B such that A G=2\ G B ,\ E\ is a point of D C such that C E=2D E\ a n d\ F is the point of B C such that B F=2F C . Prove that: a r\ (triangle \ E F C)=1/2\ a r\ (triangle \ E B F)

In a A B C ,\ \ B D and C E are the altitudes. Prove that A D B and A E C are similar. Is C D B ~ B E C ?