Online Geometry theorems, problems, solutions, and related topics.
Geometry Problem. Post your solution in the comment box below.Level: Mathematics Education, High School, Honors Geometry, College.Click the diagram below to enlarge it.
http://s14.postimg.org/7bjgzefe9/pro_1081.pngDraw lines and points per attached sketchWe have OB= 2.r , OE= r, DB= rTriangle ODF is equilateralAlong OD we have OD= OB-DB => a+ b= 2r-2a => 3a+b= 2r …. (1)Along OE we have OF=OE-FE => a+b=r-b => a+2b=r.. ( 2)Along OF we have OF=2b+4c=r ……(3)From (1) and ( 2) we have a= 3/5 r and b= r/5Replace value of b in (3) we get c= r/10
Consider the altitude of the triangle, we have3a + 4b + 4c = 3rConsider thr radius of the large circle C1, 2a + b = 5b + 4c (=R)a − 2b − 2c = 0Consider the radius of the in-circle, 3b + 4c = rSummarizing, we have 3a + 4b + 4c = 3ra − 2b − 2c = 03b + 4c = rHence, a=3r/5, b=r/5, r/10.
Easy to observe thatr = 3b + 4c,b + 2a = R = r + 2b and thus r = 2a - b,2r = b + 3a.Add the last two equations:to get 3r = 5a or a = 3r/5 Next b = 2a - r = 6r/5 - r = r/5,c = r/4 - 3b/4 = r/4 - 3r/20 = 2r/20 = r/10