Geometry Problem. Post your solution in the comments box below.

Level: Mathematics Education, High School, Honors Geometry, College.

Click the diagram below for more details.

## Friday, October 16, 2015

### Geometry Problem 1154 Sangaku Problem: Three circles and a tangent line

Labels:
circle,
common tangent,
sangaku,
tangent

Subscribe to:
Post Comments (Atom)

Let the distance between the points of tangency of the tangent line be m between circles a and c and n between circles c and b so that the corresponding distance between a and b is m+n.

ReplyDeleteWe can write 3 equations using Pythagoras

m^2 + (a-c)^2 = (a+c)^2 from which

m^2 = 4ac.,,(1)

Similarly

n^2 = 4bc ....(2)

(m+n)^2 = 4ab.....(3)

Now eliminate m and n from these 3 equations

4ac + 4bc + 8c sqrt(ab) = 4ab

Divide by 4abc

1/a + 1/b + 2/sqrt(ab) = 1/c

Take the square root of both sides obviously ignoring the negative possibility

1/sqrt a + 1/sqrt b = 1/ sqrt c

Sumith Peiris

Moratuwa

Sri Lanka