tag:blogger.com,1999:blog-6933544261975483399.post212787193490230175..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Geometry Problem 1432: Tangent Circles, Secant, Tangent Lines, Proportionality, SimilarityAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-6933544261975483399.post-13458588013357139412020-05-27T12:58:43.406-07:002020-05-27T12:58:43.406-07:00By drawing the common tangent at T we can easily d...By drawing the common tangent at T we can easily deduce that AC//BD<br /><br />C^2/d^2 = CT.CD/DT.CD = CT/DT = AT/BT (because AC//BD) = AT.AB/BT.AB = a^2/b^2 <br /><br />From which a/c = b/d<br /><br />Sumith Peiris<br />Moratuwa<br />Sri Lanka<br />Sumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-31369296205115709362019-05-08T08:02:45.694-07:002019-05-08T08:02:45.694-07:00connect AC, BD;
Through T, make common tangent lin...connect AC, BD;<br />Through T, make common tangent line.<br />It is easy to see triangle ACT similar to triangle BDT;<br />so AT/CT = BT/DT ;<br />ALSO a^2 = AT*AB; c^2 = CT*CD;<br />b^2 = BT*AB; d^2 = DT*CD;<br />SO a^2/c^2 = b^2/d^2<br />so a/c= b/d<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-78990552411326412082019-05-05T19:02:27.624-07:002019-05-05T19:02:27.624-07:00https://photos.app.goo.gl/uWKn5PxzEsWQnLBy9
Let R...https://photos.app.goo.gl/uWKn5PxzEsWQnLBy9<br /><br />Let R=TO1 and r=TO2<br />Let 2 α= angle AO1T= angle TO2B<br />In triangle AO1O2, using cosine formula we have<br />AO2^2= (R+r)^2+R^2-2R(R+r)cos(2 α)<br />And AE^2=a ^2=AO2^2-r^2<br />AE^2=2R(R+r)(1-cos2 α)<br />Replace 1-cos(2 α)=2 sin(α)^2 in above we get<br />AE^2= 4R(R+r)sin(α)^2 => a= 2.sqrt(R(R+r)).sin(α)<br />In the same way we have BG= b= 2.sqrt(r(R+r)).sin(α)<br />So a/b= sqrt(R/r)<br />And ratio of tangent from A to circle O2 and B to circle O1 is not depend on angle α<br />So c/d= sqrt(R/r) and a/b=c/d or a/c=b/d<br />Peter Tranhttps://www.blogger.com/profile/02320555389429344028noreply@blogger.com