See complete Problem 127
Triangle, Centroid, Incenter, Parallel, Proportions. Level: High School, SAT Prep, College geometry
Post your solutions or ideas in the comments.
Monday, June 23, 2008
Elearn Geometry Problem 127
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1)[AGC]=[ABC]/3=sr/3=br/2 hence b=(a+c)/2
ReplyDelete2)M midpoint of AC; bisector BI and AC meet at D
AM=b/2; DC= ab/(a+c)=a/2
triangles BGI and BMD are similar with the ratio
2/3
GI=(2/3)MD= (c-a)/6
Solution to problem 127.
ReplyDeleteLet be BS the angle bisector and BM the median. From problem 126 we know that BI/IS = (a+c)/AC. Since GI and MS are parallel, then
BI/IS = BG/GM = 2,
so (a+c)/AC = 2 and AC = (a+c)/2.
As BS is the angle bisector, we have
c/AS = a/SC = (c-a)/(AS-SC) = (c+a)/(AS+SC) = (c+a)/AC = 2.
Thus (c-a)/(AS-SC) = 2 (1).
Besides, AS – SC = AM + MS – SC = MC + MS – SC = MS + SC + MS – SC = 2MS.
As BGI and BMS are similar, MS = (3GI)/2, then AS – SC = 2.(3GI)/2 = 3GI.
From (1) we get (c-a)/(3GI) = 2 and GI = (c-1)/6.
(GAC) = (IAC)
ReplyDelete∆/3 = (1/2)(br)= b∆/2s
3b = 2s
3b = a + b + c
b = (a + c)/2
Extend GI either way to meet BA at D and BC at E.
DG = GE = DE/2 = 2AC/6 = b/3 and BI/IE = sin C / sin (B/2)
GI = GE - IE = b/3 - BI sin(B/2)/sin C = b/3 - r/sin C
= b/3 - ra/h where h is altitude from B
But r = h/3 since GI∥AC
So GI = b/3 - a/3 = (b- a)/3 = (2b - 2a)/6 = (c - a)/6