# Antiparallelogram Property

- Author:
- asifsound

cf. GeoGebra's sense is odd.
---- GeoGebra's specification of circle intersection is odd.

**■ Kempe's reversor**Line Symmetry Linkage (Alias: Imai's Butterfly) --- by 5 bars. exact straight line linkage --- by 7 bars. ( Peaucellier exact straight line linkage --- by 7 bars. )**■ Hart's inversor (= Antiparallelogram)**--- by 4 bars. exact straight line linkage --- by 5 bars.**Tip:**Peaucellier's inversor is horizontal-symmetric (6 bars), Hart's inversor is vertical-symmetric (4 bars). Element is the same triangle parts. ( Hart is a pure mathematician in UK. Peaucellier is not a pure mathematician in France. He was a French army officer. )**■ Hart's A-frame exact line straight linkage**--- by 5 bars. cf. Generalized Hart's A-frame**■ Spiral**In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.**■ Conservation law (like as "conservation law of energy")**●C a |h b | A● + + ●B M H , where AH = a', HB = b', AM = 0.5(a'+ b') = 0.5 AB, CH = h (= vertical height) then, a^{2}- b^{2}= (a'^{2}+ h^{2}) - (b'^{2}+ h^{2}) = a'^{2}- b'^{2}= (a'+ b')(a'- b') = constant (∵ a, b is constant). also, a'- b' = 2[a'- 0.5(a'+b')] = 2 MH ∽ MH a'+ b' = AB i.e.**MH × AB**= constant. ( i.e. if AB increases by N multiple, MH increases by 1/N multiple. ) This law is always true for any angle ∠ACB. Conservation law in the triangle geometry.**■ Conservation law of Parallelogram**C●---------●D / / A●---------●B if, AB = a, AC = b, AB // CD, AC // BD, Then, Q: What is AD^{2}+ CB^{2}? A:**2 (a**= constant.^{2}+ b^{2})