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

AD is the bisector of /_BAC of /_\ABC , where D is a point on BC. Prove that (BD)/(DC)=(AB)/(AC) .

l is the perpendicular bisector of the line segment PQ and R is a point on the some side of l as P. The line segment QR intersects l at X. Prove that PX + XR = QR .

If x and y are the mid-points of the sides CA and CB respectively of a ∆ABC right angled at C. prove that 4Ay^2=4AC^2 +BC^2

Find the equation of the perpendicular bisector of the line segment joining the points (7, 4) and (-1, -2).

Prove that the locus of the centres of all circles passing through two given points A, B is the perpendicular bisector of the line segment AB.

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

In the triangle ABC, the bisector of /_A intersects BC at X. XL and XM are perpendiculars from X on AB and AC respectively. Prove that AL= AM .

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 .

ABC is a right angled triangle right angled at C. CD is perpendicular on the hypotenuse AB. Prove that (BC^2)/(AC^2)=(DB)/(AD) .

Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of the bisectors of the corresponding angles of the triangles. [The end-points of the angular bisectors are on the opposite sides of the angles.]