Prove that any two sides of a triangle are together greater than twice the median drawn to the third side. GIVEN : `triangle A B C` in which `A D` is a median. PROVE : `A B+A C >2A D` CONSTRUCTION : Produce `A D` to `E` such that `A D=D E`. Join `E C`.

Prove that sum of any two sides of a triangle is greater than twice the median with respect to the third side.

Prove that the perimeter of a triangle is greater than the sum of the three medians. GIVEN : A A B C in which A D ,B E and C F are its medians. TO PROVE : A B+B C+A C > A D+B E+C F

The sum of any two sides of a triangle is greater than the third side. GIVE : A triangle A B C TO PROVE : A B+A C > B C ,A B+B C > A C and B C+A C > A B CONSTRUCTION : Produce side B A to D such that A D=A Cdot Join C D .

Show that a median of a triangle divides it into two triangles of equal area. GIVEN : A triangle A B C in which A D is the median. TO PROVE : a r(triangle A B D)=a r( triangle A D C) CONSTRUCTION : Draw A L_|_B C

Show that the sum of the three altitudes of a triangle is less than the sum of three sides of the triangle. GIVEN :triangle A B C in which A D_|_B C ,B E_|_A C and C F_|_A Bdot PROVE : A D+B E+C F < A B+B C+A C

If D is the mid-point of the hypotenuse A C of a right triangle A B C , prove that B D=1/2A C . GIVEN : A A B C in which /_B=90^0 and D is the mid-point of A Cdot TO PROVE : B D=1/2A C CONSTRUCTION Produce B D to E so that B D=D Edot Join E Cdot

In Figure, A D is the median and D E||A Bdot Prove that B E is the median.

Two triangles are congruent if two sides and the included angle of one are equal to the corresponding sides and the included angle of the other triangle. GIVEN : Two triangles A B C and D E F such that A B=D E ,A C=D F and /_A=/_D PROVE : A B C~= D E F Figure

Angles opposite to two equal sides of a triangle are equal. GIVEN : A B C in which A B=A C TO PROVE : /_C=/_B CONSTRUCTION : Draw the bisector A D of /_A which meets B C in D Figure

Angles opposite to two equal sides of a triangle are equal. GIVEN : A B C in which A B=A C TO PROVE : /_C=/_B CONSTRUCTION : Draw the bisector A D of /_A which meets B C in D Figure