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Analytic geometry
02-18-2019, 08:48 PM (This post was last modified: 02-19-2019 07:46 PM by Albert Chan.)
Post: #5
RE: Analytic geometry
(02-18-2019 12:09 PM)yangyongkang Wrote:  I will extend this question. Calculate the area enclosed by the three tangent lines of the circle.

Area under rotated coordinate match area before rotation.
So, again assume t1=0, we already know the base, find height:

x cos(t2) + y sin(t2) = 1 ; rotated line 2
x cos(t3) + y sin(t3) = 1 ; rotated line 3

Multiply 1st eqn by sin(t3), 2nd by sin(t2), and subtract:

x (cos(t2) sin(t3) - cos(t3) sin(t2)) = sin(t3) - sin(t2)
x = (sin(t3) - sin(t2)) / sin(t3 - t2) = cos(½(t3+t2)) sec(½(t3-t2))

y value not needed here, but just in case: y = sin(½(t3+t2)) sec(½(t3-t2))

height = | 1 - x |
= | (cos(½(t3-t2)) - cos(½(t3+t2))) sec(½(t3-t2)) |
= | 2 sin(t2/2) sin(t3/2) sec(½(t3-t2)) |

base = | tan(t2/2) - tan(t3/2) |
= | (sin(t2/2) cos(t3/2) - cos(t3/2) sin(t2/2)) sec(t2/2) sec(t3/2) |
= | sec(t2/2) sec(t3/2) sin(½(t3-t2)) |

ΔArea = base*height/2, and remove the t1=0 restriction:

ΔArea = | tan(½(t2-t1)) tan(½(t3-t1)) tan(½(t3-t2)) |
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Messages In This Thread
Analytic geometry - yangyongkang - 02-18-2019, 09:27 AM
RE: Analytic geometry - yangyongkang - 02-18-2019, 12:09 PM
RE: Analytic geometry - Albert Chan - 02-18-2019 08:48 PM
RE: Analytic geometry - Albert Chan - 02-18-2019, 03:16 PM
RE: Analytic geometry - Albert Chan - 02-18-2019, 05:51 PM
RE: Analytic geometry - Albert Chan - 02-19-2019, 10:23 PM
RE: Analytic geometry - Albert Chan - 02-20-2019, 09:33 PM



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