tag:blogger.com,1999:blog-6933544261975483399.post3860255558384842853..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Problem 1103: Right Triangle, Incircle, Inscribed Circle, Radius, Geometric Mean, Sangaku, Japanese, Metric RelationsAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-6933544261975483399.post-80653359874158658152015-11-23T01:00:43.961-08:002015-11-23T01:00:43.961-08:00FODB is a square of side r so easily
r = (c+a-b)...FODB is a square of side r so easily <br /><br />r = (c+a-b)/2.......(1)<br />From similar Tr. s (r-r1)/r = r1/(c-r)......(2)<br /><br />Eliminating r we have <br /><br />r1 = (c+a-b)/(c+b-a)/4c = {c^2 - (a-b)^2}/4c<br /><br />Simplifying using c^ 2 = b^ 2 - a^2 we have<br /><br />r1 = a(b-a)/2c and similarly<br />r2 = c(b-c)/2a<br /><br />So 2r1r2 = (b-a) (b-c)/2.....(3)<br /><br />Now from (1), <br />r ^2 = (c+a-b)^2 / 4 which simplifies to (b-a)(b-c) /2 which from (3) is therefore 2r1r2<br /><br />Sumith Peiris<br />Moratuwa <br />Sri LankaSumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-78825409916397717652015-03-27T10:42:11.437-07:002015-03-27T10:42:11.437-07:00A/2 + C/2 = Pi/4,
1 + tan C/2 = 1 + tan ( Pi/4...A/2 + C/2 = Pi/4, <br />1 + tan C/2 = 1 + tan ( Pi/4 - A/2) = 1 + [(1 - tan A/2) /(1 + tan A/2)] = 2/(1 + tan A/2)<br />Let L, M be the projections of O1, O2 on AC resp..<br />C, O2, O are collinear and r/r2 = EC/ MC = 1 + EM/MC = 1 + O2M/MC = 1 + tan C/2 = 2/(1 + tan A/2)<br />A, O1, O are collinear and r/r1 = AE/AL = 1 + LE/AL = 1 + r1/AL = (1 + tan A/2)<br />Hence (r/r2)(r/r1) = 2, r^2 = 2.r1.r2Pravinhttps://www.blogger.com/profile/05947303919973968861noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-78405445249315639112015-03-25T18:54:44.236-07:002015-03-25T18:54:44.236-07:00http://s25.postimg.org/815s5zg3j/pro_1103.png
Draw...http://s25.postimg.org/815s5zg3j/pro_1103.png<br />Draw lines per sketch<br />We have ∠ (OED1)= ∠ (OEO2)= ∠ (OFO1)= ∠ (ODO2)= 45 degrees<br />And ∠ (O1FD)= ∠ (FDO2)= ∠ (O1EO2)= 90 degrees<br />Calculate O1O2^2= O1E^2+O2E^2= 2. r1^2+ 2.r2^2<br />O1O2^2= FD^2+(DO2-FO1)^2<br />Replace FD^2= 2.r^2 , DO2= r2.sqrt(2) , FO1= r1.sqrt(2) in above expression<br />We get r^2=2.r1.r2<br />Peter Tranhttps://www.blogger.com/profile/02320555389429344028noreply@blogger.com