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.