tag:blogger.com,1999:blog-6933544261975483399.post7571805157979962890..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Geometry Problem 1154 Sangaku Problem: Three circles and a tangent lineAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-6933544261975483399.post-48736184847878496472015-10-16T09:36:44.478-07:002015-10-16T09:36:44.478-07:00Let the distance between the points of tangency of...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.<br /><br />We can write 3 equations using Pythagoras <br /><br />m^2 + (a-c)^2 = (a+c)^2 from which <br />m^2 = 4ac.,,(1)<br />Similarly <br />n^2 = 4bc ....(2)<br />(m+n)^2 = 4ab.....(3)<br /><br />Now eliminate m and n from these 3 equations <br /><br />4ac + 4bc + 8c sqrt(ab) = 4ab<br /><br />Divide by 4abc<br />1/a + 1/b + 2/sqrt(ab) = 1/c<br />Take the square root of both sides obviously ignoring the negative possibility<br /><br />1/sqrt a + 1/sqrt b = 1/ sqrt c<br /><br />Sumith Peiris<br />Moratuwa<br />Sri Lanka<br />Sumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.com