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 .

Step 1: Observe the given figure and make a note of an important points
Step 2: Find the right angle triangles in the given picture and apply the Pythagoras theorem.
EXAMPLE: ΔABC is a right triangle, so by the Pythagorean theorem,
AB^2+BC^2 = AC^2 ..........................(1)
ΔABE is a right triangle
so, AB^2+BE^2 = AE^2...........................(2)
ΔDBC is a right triangle
So, DB^2+BC^2 = CD^2............................(3)
ΔDBE is a right triangle
So, DB^2 + BE^2 = DE^2...............................(4)
Step 3: Add BE^2 + DB^2 to both sides of equation (1)
EXAMPLE: AB^2 + BC^2 + BE^2 + DB^2 = AC^2 + BE^2 + DB^2
Rearrange the terms:
AB^2 + BE^2 + DB^2 + BC^2 = AC^2 + DB^2 + BE^2
Let's put parentheses
(AB^2 + BE^2) + (DB^2 + BC^2) = AC^2+(DB^2+BE^2)
AE^2 + CD^2 = AC^2 + DE^2
[ \because Equation (2), (3) and (4)]