Toolkit 41

Dual Pythagorean Theorem

CB^2-CA^2=EB^2-EA^2=FB^2-FA^2=DB^2-DA^2

Proof

By applying the Pythagorean Theorem to triangles $\triangle BCD$ and $\triangle ACD$ ,

CB^2 = CD^2 + DB^2,

CA^2 = CD^2 + DA^2.

Therefore,

CB^2 - CA^2 = DB^2 - DA^2.

Similarly,

EB^2 - EA^2 = DB^2 - DA^2,

FB^2 - FA^2 = DB^2 - DA^2.

Hence,

CB^2 - CA^2 = EB^2 - EA^2 = FB^2 - FA^2 = DB^2 - DA^2. \quad\square