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Toolkit 41

Dual Pythagorean Theorem

Dual Pythagorean Theorem figure
CB2−CA2=EB2−EA2=FB2−FA2=DB2−DA2CB^2-CA^2=EB^2-EA^2=FB^2-FA^2=DB^2-DA^2

Proof

Dual Pythagorean Theorem figure

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

CB2=CD2+DB2,CB^2 = CD^2 + DB^2,
CA2=CD2+DA2.CA^2 = CD^2 + DA^2.

Therefore,

CB2−CA2=DB2−DA2.CB^2 - CA^2 = DB^2 - DA^2.

Similarly,

EB2−EA2=DB2−DA2,EB^2 - EA^2 = DB^2 - DA^2,
FB2−FA2=DB2−DA2.FB^2 - FA^2 = DB^2 - DA^2.

Hence,

CB2−CA2=EB2−EA2=FB2−FA2=DB2−DA2.□CB^2 - CA^2 = EB^2 - EA^2 = FB^2 - FA^2 = DB^2 - DA^2. \quad\square