Analytic geometry

02192019, 10:23 PM
(This post was last modified: 02202019 09:48 PM by Albert Chan.)
Post: #6




RE: Analytic geometry
Trivia: If triangle inscribed unit circle, Δarea = Δhalfperimeter
Prove: Again, assume t1=0, and normalized t2, t3, such that 2Pi > t3 > t2 > 0 To have unit circle inside triangle require these conditions: 0 < t2 < Pi ; made triangle angle Pi  t2 Pi < t3 < Pi + t2 ; made triangle angle t3  Pi > tan(t2/2) > 0, tan(t3/2) < 0, tan((t3t2)/2) > 0 a =  tan(t3/2)  tan(t2/2)  = tan(t2/2)  tan(t3/2) b =  tan(t2/2)  tan(½(t3t2))  = tan(t2/2) + tan(½(t3t2)) c =  tan(t3/2)  tan(½(t2t3))  = tan(t3/2) + tan(½(t3t2)) s = ½(a + b + c) = tan(t2/2)  tan(t3/2) + tan(½(t3t2)) = tan(½(t3t2)) * (1  (1 + tan(t2/2) tan(t3/2))) =  tan(t2/2) tan(t3/2) tan(½(t3t2)) Add back absolute function to remove sign, and remove t1=0 restriction: s = tan(½(t2t1)) + tan(½(t3t1)) + tan(½(t3t2)) = tan(½(t2t1)) tan(½(t3t1)) tan(½(t3t2)) Match previously derived Δarea formula. QED Comment: tan(...) peices are length of circle tangents to triangle vertice. 

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Messages In This Thread 
Analytic geometry  yangyongkang  02182019, 09:27 AM
RE: Analytic geometry  yangyongkang  02182019, 12:09 PM
RE: Analytic geometry  Albert Chan  02182019, 08:48 PM
RE: Analytic geometry  Albert Chan  02182019, 03:16 PM
RE: Analytic geometry  Albert Chan  02182019, 05:51 PM
RE: Analytic geometry  Albert Chan  02192019 10:23 PM
RE: Analytic geometry  Albert Chan  02202019, 09:33 PM

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