Two sides AB, BC and median AM of one triangle ABC are respectively equal to sides PQ and QR and median PN of `DeltaPQR`. Show that:
`(i) DeltaABM ~= DeltaPQN`
`(ii) DeltaABC ~= DeltaPQR`

In a triangle ABC (see figure), E is the midpoint of median AD, show that (i) ar DeltaABE = ar DeltaACE (ii) ar DeltaABE=1/4ar(DeltaABC)

E and F are respectively the mid-points of equal sides AB and AC of DeltaABC (see figure) Show that BF = CE.

ABC is a triangle in which altitudes BD and CE to sides AC and AB are equal (see figure) . Show that (i) DeltaABD ~= DeltaACE (ii) AB = AC i.e., ABC is an isosceles triangle.

If the median AD of triangle ABC makes an angle pi/4 with the side BC, then find the value of |cotB-cotC|dot

BE and CF are perpendicular on sides AC and AB of triangles ABC respectively . Prove that four points B , C ,E , F are concyclic. Also prove that the two angles of each of DeltaAEFandDeltaABC are equal.

In Delta^(s)ABC and, AB"||"DE, AB=DE, BC=EF and BC"||"EF . Vertices A, B and C are joined to vertices D, E and F respectively (see figure). Show that (i) ABED is a parallelogram (ii) BCFE is a parallelogram (iii) AC = DF (iv) DeltaABC~=DeltaDEF

If AD and PM be two medians of DeltaABC and DeltaPQR respectively, where DeltaABC~DeltaPQR , then prove that (AB)/(PQ)=(AD)/(PM) .

Let us consider one vertex and one side through the vertex along x -axis of a triangle . Now the coordinates of the vertices B,C and A of any triangle ABC (0,0) ,(a,0) and (h,k) respectively should be taken . If in triangle ABC , AC =3 ,BC=4 and median AD is perpendicular with median BE , then the area of DeltaABC is -

Two triangles ABC and ABD of equal areas are on the opposite sides of AB. Prove that AB bisects CD equally.