Curved Lines in a Circle

* If angle x is in radian, curved side BKC=ax & curved side DKE=ax1. ..........(1)

* BC=2a*sin(x/2) and DE=2a*sin(x1/2) .....(1a)                          Since BC2=a2+a2-2a2cos x

* BkC/BC=(x/2)/sin (x/2)  and DkE/DE=(x1/2)/sin (x1/2) .........(1b)

    As x -> 0, the RHS tends to 1 and Bkc=BC and DkE=DE

* AL/KL= a*cos (x/2)/[a*(1-cos (x/2)]=1/[1/cos(x/2)  -1]  and AT/KT= 1/[1/cos(x1/2)  -1] ......(1c)

* Perimeter of the curved triangle ABkC is a(2+x) & curved triangle ADkE=a(2+x1) ........(2)

* Area of the curved triangle ABkC is (π*a2/2π)*x = a2x/2=(1/2)*ax*a .........(3)

* Area of the curved triangle ADkE is (π*a2/2π)*x1 = a2x1/2=(1/2)*ax1*a ........(4)

* Analogous to a triangle ABC where a/(b+c-a) +b/(a+c-b)+c/(a+b-c) =z which has a minimum value 3, here

   z=2/ x  +x/(2-x)  for Curved triangle ABkC ........(5)

  z1=2/ x1  +x1/(2-x1)  for Curved triangle ADkE ......(6)

* Triangular inequality : a+ax >a  => ax > 0 which is true  .....(7)

* Triangular inequality : a+a >ax  may or may not hold good unlike a st. line triangle where it invariably holds good.

* Putting the cosine law, a2x2=a2+a2-2a2cos Y where Y is a hypothetical angle, we get Y=2*sin-1(x/2)

* Similarly, Putting the cosine law, a2=a2x2+a2-2x2a2cos U where U is a hypothetical angle, we get U=cos-1(x/2)

* In curved triangle ABkC, sum of angles Y+2U=  2*sin-1(x/2)  +2*cos-1(x/2)    = 2*π/2 = π where Y,U are hypothetical angles. Same for the other curved triangle.

While analysing the equations of curved triangles, one finds that the area formula is equivalent to that of straight triangle.The triangular inequality partly holds good. Cosine formula can be fitted in with hypothetical angles etc.

* If the slope of AE is m2', then the x & y co-ordinate of E (m2,n2) are related by formula m2m2' -g1m2'+h1=n2 ; and m2 can be found out by solving the quadratic equation (1+m2'*m2')m2*m2 +2g1(m2'*m2' -1)m2 +(g1*g1+g1*g1*m2'*m2'-a*a)=0. since m2' has 2 values and for each value of m2', m2 has 2 values => m2 has total 4 values and n2 has 4*2=8 values since m2' has 2 values.

Value of a
Value of x in Degree
Value of x1 in Degree
Data below is independent of data above  
x-coordinate of center at A(g1)
y-coordinate of center at A(h1)
x-coordinate of D-put any arbitrary value (m1)
x in radian
Comment on x
x1 in radian
Comment on x1
sum of 2 sides/3rd side(2a/ax) in ABkC
sum of 2 sides/3rd side(2a/ax1) in ADkE
Perimeter of ABkC-P1 [a(2+x)]
Perimeter of ADkE-P2 [a(2+x1)]
Area of ABkC-A1
Area of ADkE-A2
Angle Y
Angle U
Y+2U in degree
Angle Y1
Angle U1
Y1+2U1 in degree
If BC becomes the new arc, corresponding angle x2 is (radian) AND (degree)
If BC becomes the new arc of the circle, % change in angle x is(-Ve) % (this is independent of magnitude of radius)
General Converter    
Put angle in Radians
Find angle in Degree
Equation of the circle (x-  )2 +(y-)2 = =
or simplified equation of circle x2 + y-  () x - () y + () = 0
y-coordinate of D-(n1a) & (n1b) =± √[a2-(m1-g1)2] +h1   and 
Slope of AD w.r.t.  x-axis   and 
Slope of AE w.r.t.  x-axis   and 
x-coordinate of E (m2) , , and
y-coordinate of E (n2) , , and
y-coordinate of E (n2) , , and