* 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
BC^{2}=a^{2}+a^{2}-2a^{2}cos 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*a^{2}/2π)*x =
a^{2}x/2=(1/2)*ax*a .........(3) * Area of the curved triangle ADkE is (π*a^{2}/2π)*x1 =
a^{2}x1/2=(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**, a^{2}x^{2}=a^{2}+a^{2}-2a^{2}cos Y
where Y is a hypothetical angle, we get Y=2*sin^{-1}(x/2) *
Similarly, Putting the **cosine
la**w, a^{2}=a^{2}x^{2}+a^{2}-2x^{2}a^{2}cos
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.
Explanation: Suppose slope of AE is m2, Co-ordinate of E is (x2,y2),
then m2=(y2-h1)/(x2-g1) or y2=m2x2+(h1-g1m2)=m2x2+k where k=h1-g1m2
or y2=m2x2+k.......(5)
Now, (g1-x2)^{2} +(h1-y2)^{2} =a^{2} or
(g1-x2)^{2} +(h1-k-m2x2)^{2} =a^{2} or
(g1-x2)^{2} +(g1m2-m2x2)^{2} =a^{2} or
on simplification, we get
(1+m2^{2} )x2^{2} + (-2g1)(1+m2^{2}
)x2 + (g1^{2}m2^{2} +g1^{2} - a^{2} ) =0
which is a quadratic equn. which gives 2 value of x2 for each value
of m and hence we get in total 4 values of x2.
* 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 .
* TZ=a*sin(x1r/2)*tan(x1r/2) and AZ=AT+TZ=a / Cos(x1r/2).
Previously, TZ was written as [(a*cosx1)/(2*cos(x1/2))] which is
wrong. pl. check.
* Value of AT, TZ found out from length and from co-ordinates tally
for angle > 0 and up to 162.54 degree and at 270 degree. From
co-ordinates , we take AT1 and AT8 for AT and TZ1 & TZ4 for TZ.
Reasons not yet explored. |