tag:blogger.com,1999:blog-6933544261975483399.post7326440791541997705..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Geometry Problem 1388: Triangle 40-60-80 degree, Incenter, CongruenceAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-6933544261975483399.post-91576651339564658672023-03-22T15:44:43.416-07:002023-03-22T15:44:43.416-07:00https://photos.app.goo.gl/M6AHCEaycVHP1j7v5https://photos.app.goo.gl/M6AHCEaycVHP1j7v5c.t.e.o.noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-79915515370724605352018-09-27T10:30:42.317-07:002018-09-27T10:30:42.317-07:00Considering usual triangle notations,
let BI inte...Considering usual triangle notations, <br />let BI intersect AC at D and CI intersect BA at E<br />We know AD=bc/a+c and DC=ab/a+c (Angle bisetor theorem)<br />Observe that BEC is isosceles and EC=a<br />Similarly BD is isosceles and BD=DC=ab/a+c --------(1)<br /><br />Since BDA similar to CBA (AAA)<br />=> BD=CB.AB/AC<br />=> BD=a.c/b-----------(2)<br />(1)=(2)<br />=>c=b^2/(a+c) ------------(3)<br /><br />Observe that DIC is similar to EAC<br />=>IC=AC.DC/EC<br />=>IC=b.(ab/a+c)/a (substitute values for DC and EC)<br />=>IC=b^2/(a+c) ------------(4)<br /><br />From (3) & (4) the result followsSailendra Thttps://www.blogger.com/profile/12056621729673423024noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-65495198692042629192018-09-26T09:29:55.357-07:002018-09-26T09:29:55.357-07:00Let D be the intersection point of the continuatio...Let D be the intersection point of the continuation of IC on AB, E be the intersection of the continuation of BI on AC and let x = IC and y = EC.<br /><br />1. tr ADC is similar to tr IEC so let k be the scale factor and CD = ky, AC = kx<br />2. Tr CBD is isosceles (40-80-80) so BC = CD = ky<br />3. Tr BEC is also isosceles (40-100-40) so EC = BE = y<br />4. Tr ABC is similar to AEC (40-60-80) so AB/AC = BE/BC = y/ky = 1/k<br /><br />which implies AB = AC * 1/k = kx * 1/k = x or in other words AB = IC = x.<br /><br />Benjamin Leishttps://www.blogger.com/profile/10974191081762367425noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-91799683388415193342018-09-25T17:49:58.218-07:002018-09-25T17:49:58.218-07:00Make AD = AB, D on AC;
so ABD is equilateral trian...Make AD = AB, D on AC;<br />so ABD is equilateral triangle;<br />so angle DBC=20; angle BDC =120 = angle CIB,<br />so triangle BDC equal to triangle BIC;<br />so CI=BD=AB=cAnonymousnoreply@blogger.com