The perpendicular AD to the side BC of `/_\ABC` is such that BD : CD : AD = 2:3:6. Find `angleBAC`

The perpendicular from A on side BC of a triangleABC intersects BC at D such that DB = 3 CD (see Fig. 6.55). Prove that 2AB^2=2AC^2+BC^2 .

The perpendicular from C on the side AB of a triangle ABC intersect AB at D and BD=mAD . Prove that (m+1)BC^2=(m+1)AC^2+(m-1)AB^2 .

O is a point in the interior of ∆ABC . OD,OE and OF are the perpendicular drawn to the sides BC, CA and AB respectively. Prove that AF^2+BD^2+CE^2=AE^2+BF^2+CD^2 .

D and E are respectively two points on the sides AB and AC of triangleABC such that DE||BC .If (AD)/(DB)=2/3 find (BC)/(DE)

A and D are two points on the perpendicular bisector of BC lying on the same side of BC. Prove that /_ABD = /_ACD .

D is the point on the side BC of ABC such that angleADC = angleBAC . Prove that, (BC)/(CA) = (CA)/(CD) .

D and E are respectively the point on the sides AB and AC of a trinagle ABC such that DE|\|BC . Through the point E a line parallel to CD is drawn which cut AB at the point F then show that AD^2=ABxxAF

A circle is touching the side BC of /_\ABC at P and touching AB and AC produced at Q and R are respectively. Prove that AQ=1/2(AB+BC+CA)=1/2(Perimeter of /_\ABC) .

D is a point on the side BC of a triangle ABC such that angleADC=angleBAC .Show that CA^2=CB.CD .