Click the figure below to view the Kosnita's Theorem: Circumcenters, Concurrent lines.
See more:
Kosnita's Theorem: Circumcenters
Level: High School, SAT Prep, College geometry
Thursday, November 26, 2009
Kosnita's Theorem: Circumcenters, Concurrent lines
Subscribe to:
Post Comments (Atom)
∠AOB=2∠ACB=2∠C
ReplyDelete∠AOCB=2(180°-∠AOB)=360°-4∠C
∠OCAB=(180°-∠AOCB)/2=2∠C-90°
Let C' be the point of intersection of AB and COC.
Area of ΔCAOC=1/2×AC×AOC×sin(A+∠OCAB)=1/2×AC×AOC×sin(A+2C-90°)=1/2×AC×AOC×sin(90°+C-B)=1/2×AC×AOC×cos(C-B)
AC'/C'B=Area of ΔCAOC/Area of ΔCBOC=1/2×AC×AOC×cos(C-B)/(1/2×BC×AOC×cos(C-A))=AC×cos(C-B)/(BC×cos(C-A))
AC'/C'B×BA'/A'C×CB'/B'A=1