In an isosceles triangle `A B C` with `A B=A C` , `B D` is perpendicular from `B` to the side `A C` . Prove that `B D^2-C D^2=2\ C D A D`

In Fig. 4.124, E is a point on side C B produced of an isosceles triangle A B C with A B=A C . If A D_|_B C and E F_|_A C , prove that (i) A B D E C F (ii) A BxxE F=A DxxE C . (FIGURE)

In Figure, A B C is an isosceles triangle in which A B=A CdotC P A B and A P is the bisector of exterior /_C A D of A B C . Prove that /_P A C=/_B C A and (ii) A B C P is a parallelogram.

If A B C is an isosceles triangle such that A B=A C and A D is an altitude from A on B C . Prove that (i) /_B=/_C (ii) A D bisects B C (iii) A D bisects /_A

In A B C , A D is a median. Prove that A B^2+A C^2=2\ A D^2+2\ D C^2 .

Fill in the blanks in the following so that each of the following statements is true Sides opposite to equal angles of a triangle are ....... Angle opposite to equal sides of a triangle are ....... In an opposite to equal sides of a triangle are ....... In a A B C if /_A=/_C , then A B= ..... If altitudes C E a n d B F of a triangle A B C are equal, then A B= ........ 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 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 is a triangle. D is the mid point of B Cdot If A D is perpendicular to A C , then prove that cos A\ dotcos C=(2(c^2-a^2))/(3a c) .

Delta ABC is an isosceles triangle in which A B= A C . Side BA is produced to D such that A D = A B (see Fig. 7.34). Show that /_B C D is a right angle.

In a quadrilateral A B C D , given that /_A+/_D=90o . Prove that A C^2+B D^2=A D^2+B C^2 .

A B C is an isosceles triangle such that A B=A C a n d A D is the median to base B C . Then, /_B A D= 55^0 (b) 70^0 (c) 35^0 (d) 110^0

A B C\ a n d\ D B C are two isosceles triangles on the same bas B C and vertices A\ a n d\ D are on the same side of B C . If A D is extended to intersect B C at P , show that A P bisects "\ "/_A as well as /_D A P is the perpendicular bisector of B C