On the sides of a given triangle ABC, draw the squares ABDE, BCFG and ACHK.

Prove that the triangle BDG has same area as ABC. Hint

We may even go further and draw square DGMN on DG, all the blue triangles have the same area as ABC.

Proof the "Pythagora like" relation : a² + c² = (b² + d²)/2.
Hint

What becomes this relation if ABC is right angled at B ?
Solution

Relation on sides of ABC for NEK to be in line ? Hint