In `Delta ABC, AD_|_ BC `. Prove that`AB^(2) - BD^(2) =AC^(2) -CD^(2)`

If in DeltaABC, AD_|_ BC, then prove that AB^(2) + CD^(2) = AC^(2) + BD^(2)

ABC is a right triangle right angled at B. Let D and E be any points on AB and BC respectively. Prove that AE^(2) + CD^(2) = AC^(2) + DE^(2) .

ABC is an equilateral triangle. AD is perpendicular to BC. Prove that AB^(2) + BC^(2) + CA^(2) =4AD^(2)

In Delta ABC , AD is a median and AM _|_ BC.Prove that (a) AC^(2) = AD^(2) +BC.DM+ ((BC)/(2))^(2) (b) AB^(2) = AD^(2) -BC.DM+ ((BC)/(2))^(2) (c ) AC^(2) + AB^(2) = 2AD^(2) +(1)/(2) BC^(2)

A perpendicular AD is drawn on BC from the vertex A of the acute Delta ABC . Prove that AC^(2) = AB^(2) +BC^(2) -2BC.BD OR Prove that the square drawn on the opposite side of theacute angle of an acute triangle is equal to the areas of the sum of the squares drawn on its other two sides being subtracted by twice the area of the rectangle formed by one of its sides and the projection of the other side to this side.

Two perpendicular BD and CE are drawn from the vertices B and C respectively on the sides AC and AB of the Delta ABC ,which intersect each other at the point P. Prove that AC^(2) + BP^(2) = AB^(2) + CP^(2)

In the right-angled ABC, /_A =1 right angle. BE and CF are two medians of Delta ABC . Prove that 4 ( BE^(2) + CF^(2)) =5 BC^(2)

ABCD is a rhombus.Prove that AB^(2) + BC^(2) +CD^(2) + DA^(2) = AC^(2) + BD^(2)

ABC is a right-angled isoscels triangles of which /_C is a right angle. If D is any point on AB. Then prove that AD^(2) + BD^(2) = 2CD^(2)

In Delta ABC , AD_|_ BC which intersects BC at D. If BD =3 CD , then prove that 2AB^(2) = 2AC^(2)+BC^(2)