tag:blogger.com,1999:blog-6933544261975483399.post8052036148786358972..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Geometry Problem 1157: Four Circles, Mutually Tangent, 90 Degrees, Radius, Metric RelationsAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-6933544261975483399.post-10548834625447940052016-01-12T09:39:58.248-08:002016-01-12T09:39:58.248-08:00For some reason as yet unknown to me < O1O4O3 =...For some reason as yet unknown to me < O1O4O3 = 90<br /><br />Then using Pythagoras for Tr. O1O4O3 we get a quadratic from which r4 = 20/3<br /><br />Or easier we can use Ptolemy on cyclic quadrilateral O1O4O3O2<br /><br />But the 1st step is still beyond me.<br /><br />Ajit / Antonio - any ideas? Sumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-24924730900982796632015-10-28T01:15:46.849-07:002015-10-28T01:15:46.849-07:00(To Ajit)
Can u pls explain why r4=r1+r2+r3 ?
(To Ajit)<br />Can u pls explain why r4=r1+r2+r3 ?<br />Anonymoushttps://www.blogger.com/profile/05179313392359870009noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-33492726668604282152015-10-28T01:12:58.359-07:002015-10-28T01:12:58.359-07:00From right trO1O2O3, r3=5/3. Let <O1O2O4=x. Usi...From right trO1O2O3, r3=5/3. Let <O1O2O4=x. Using cosine rule in trs O1O2O4 & O3O2O4, and using the fact sin^2x+cos^2x=1 we get a quadratic in r4 which yeilds r4=20/3.Anonymoushttps://www.blogger.com/profile/05179313392359870009noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-12110289868208656742015-10-26T21:06:40.564-07:002015-10-26T21:06:40.564-07:00Sumith,
Please check this out: http://euler.genepe...Sumith,<br />Please check this out: http://euler.genepeer.com/from-herons-formula-to-descartes-circle-theorem/<br />and also: http://www.had2know.com/academics/descartes-theorem-soddy-circle-calculator.html<br />If you find any proof simpler than this, you can write to me at: ajitathle@gmail.comAjithttps://www.blogger.com/profile/00611759721780927573noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-45987227505734567992015-10-26T19:03:39.134-07:002015-10-26T19:03:39.134-07:00Can u share the basic steps of the proof ie for th...Can u share the basic steps of the proof ie for the formulae of Rext and Rint Ajit ?<br /><br />Rgds<br /><br />SumithSumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-75902281480226090562015-10-25T19:23:32.379-07:002015-10-25T19:23:32.379-07:00If we have three touching circle of radii a, b &am...If we have three touching circle of radii a, b & c then the radius of the external circle that touches all three may be obtained from, Rext = abc/(ab+bc+ca -2√abc(a+b+c)), (The relevant problem # from this collection which gives this formula now eludes me)<br />In this case, a=4,b=1 & c=5/3. Apply the formula above to obtain, r4 =20/3. The radius of the internal circle which touches the inside three circles may be obtained from: Rint = abc/(ab+bc+ca +2√abc(a+b+c)),Ajithttps://www.blogger.com/profile/00611759721780927573noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-33790400605868885692015-10-25T18:31:20.545-07:002015-10-25T18:31:20.545-07:00Got easily r3 = 5/3
But r4 still eludes meGot easily r3 = 5/3<br /><br />But r4 still eludes meSumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-15426034211785667842015-10-25T18:11:43.441-07:002015-10-25T18:11:43.441-07:00From rt. triangle, O1O2O3, (4+1)²+(1+r3)²=(4+r3)² ...From rt. triangle, O1O2O3, (4+1)²+(1+r3)²=(4+r3)² or r3 = 5/3 and thus r4 = 4+5/3+1=20/3.Ajithttps://www.blogger.com/profile/00611759721780927573noreply@blogger.com