Geometry Problem. Post your solution in the comments box below.
Level: Mathematics Education, High School, Honors Geometry, College.
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Friday, October 16, 2015
Geometry Problem 1154 Sangaku Problem: Three circles and a tangent line
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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