## Monday, February 9, 2015

### Geometry Problem 1081: Equilateral Triangle, Inscribed Circle, Inradius, Tangent Circles, Radius, Tangent Line, Sangaku Japanese Problem

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. 1. http://s14.postimg.org/7bjgzefe9/pro_1081.png
Draw lines and points per attached sketch
We have OB= 2.r , OE= r, DB= r
Triangle ODF is equilateral
Along 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/5
Replace value of b in (3) we get c= r/10

2. Consider the altitude of the triangle, we have
3a + 4b + 4c = 3r

Consider thr radius of the large circle C1,
2a + b = 5b + 4c (=R)
a − 2b − 2c = 0

Consider the radius of the in-circle,
3b + 4c = r

Summarizing, we have
3a + 4b + 4c = 3r
a − 2b − 2c = 0
3b + 4c = r

Hence,
a=3r/5, b=r/5, r/10.

3. Easy to observe that
r = 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