Geometry Problem
Click the figure below to see the complete problem 447 about Complete Quadrilateral, Inradii, Exradii, Sides.
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Complete Problem 447
Level: High School, SAT Prep, College geometry
Friday, May 7, 2010
Problem 447: Complete Quadrilateral, Inradii, Exradii, Sides.
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E, F, G, H tg points of AB, BC, CD, AD
ReplyDelete▲EOaA ~ ▲AHOd, ▲EOaB ~ ▲BFOb (OaOd bisector of kites)
=>
AE/ra = AH/rd, EB/ra = BF/rb =>
AE/ra + EB/ra = AH/rd + BF/rb
a/ra = AH/rd + BF/rb (1)
▲FObC ~ ▲CGOc, ▲COcD ~ ▲DHOd
FC/rb = CG/rc, GD/rc = DH/rd =>
CG/rc + GD/rc = FC/rb + DH/rd
c/rc = FC/rb + DH/rd (2)
from (1) and (2)
a/ra + c/rc = BF/rb + FC/rb + AH/rd + DH/rd
A/ra + c/rc = b/rb + d/rd
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Let external angle of ABCD at A,B,C,D be A',B',C',D'
ReplyDeleteLet O be the excenter of ABE, excircle of ABE touch AB at P, then
a/r_a=AP/r_a+PB/r_a=cot(A'/2)+cot(B'/2)
similarly c/r_c=cot(C'/2)+cot(D'/2)
and a/r_a+c/r_c=b/r_b+d/r_d=cot(A'/2)+cot(B'/2)+cot(C'/2)+cot(D'/2) is obvious