# 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, a2 - b2 = (a'2 + h2) - (b'2 + h2) = 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 AD2 + CB2 ? A: 2 (a2 + b2) = constant.