tag:blogger.com,1999:blog-6500033298667240354.post4497071426297959516..comments2024-02-19T00:34:12.578-08:00Comments on Always Creative Geometry Problems plus Occasionally Annoying Philosophy: Fox 2938foxeshttp://www.blogger.com/profile/09567328431908997738noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-6500033298667240354.post-66343874118645924772010-06-23T00:45:37.152-07:002010-06-23T00:45:37.152-07:00ok. i was missing the DU=DA part. thanks
bleaugok. i was missing the DU=DA part. thanks<br /><br />bleaugAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-82674589964040390852010-06-22T08:25:33.522-07:002010-06-22T08:25:33.522-07:00Here is probably the simpliest solution:
(Name the...Here is probably the simpliest solution:<br />(Name the rectangle ABCD starting with the top left hand corner and going anti-clockwise - As Ajit defined)<br /><br />1. |AD|=|UD|=2e (they are the common tangents from the same point).<br />2. Let DU intersect BC in point P.<br />3. |UP|=|BP|=e (they are the common tangents from the same point).<br />4. |PC| = 2e-e = e (ABCD is a rectangle).<br />5. Connect point U to the center of the circle, point O.<br />6. m(/_UPC)=a (Try to see the identical deltoids (kites) sharing the same side OU.<br />7. cos(a)=cos(/_UPC) = e/3e = 1/3.<br /><br />-binaryAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-77956022884251337302010-06-22T08:23:21.971-07:002010-06-22T08:23:21.971-07:00BE =EU (equal tangents) and DU =DA (equal tangents...BE =EU (equal tangents) and DU =DA (equal tangents). Since EU=UH=HD we, therefore, can say that EC = a/2 and ED=3a/2. <br />What's the difficulty here?<br />AjitAjithttps://www.blogger.com/profile/00611759721780927573noreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-16989478724675622062010-06-22T02:36:10.879-07:002010-06-22T02:36:10.879-07:00your reasoning is indeed correct but you seem to c...your reasoning is indeed correct but you seem to consider BE=EC=a/2 as obvious. I can't find a decisive argument to support that, except by assuming cos(DEC)=1/3 which obviously goes into circles... Any better idea?<br /><br />bleaugAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-25399057414753546682010-06-21T23:53:08.574-07:002010-06-21T23:53:08.574-07:00One easier way of doing this is as follows: Name t...One easier way of doing this is as follows: Name the rectangle ABCD starting with the top left hand corner and going anti-clockwise. Let AB=2b and BC=a as before. Let DU meet BC in E and let midpt. of UD be F. Now BE = EU = UF =FD = a/2 and thus EC=a/2 but DE = 3a/2 and hence cos(DEC)= 1/3. Now if O be the centre of the semi-circle then triangles OBE, OEU, OUF & OFT are all congruent and thus α = /_TFD = /_BOU = /_DEC. Hence etc. Wud that be correct?<br />AjitAjithttps://www.blogger.com/profile/00611759721780927573noreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-48408091029944546472010-06-20T17:04:29.976-07:002010-06-20T17:04:29.976-07:00Yes, indeed. cos(α)= 1/3. I made an error in calcu...Yes, indeed. cos(α)= 1/3. I made an error in calculation. Here's the complete solution: Let the semi-circle be x^2+y^2=b^2 with the right top corner as (a,b). The tangent is ax1+by1=b^2 with U as (x1,y1). So x1^2+y1^2=b^2 which gives y1=b(b^2-a^2)/(b^2+a^2) = -(b-2b/3) giving us a=b√2. Now let 180-α=2Φ. From the figure,tan(Φ)=b/(b√2/2)= √2 or (sec(Φ))^2=1+2=3 which in turn gives (cos(Φ))^2=1/3 or cos(2Φ)=2/3 -1=-1/3 and hence cos(α) = 1/3<br />AjitAjithttps://www.blogger.com/profile/00611759721780927573noreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-2088183385619180562010-06-20T14:11:29.407-07:002010-06-20T14:11:29.407-07:00We have 2 independent solutions, stating that the ...We have 2 independent solutions, stating that the answer is 1/3. So 1/9 must be wrong. We should correct the question later. Sorry for the trouble.<br />- 8foxesAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6500033298667240354.post-47827805242185637282010-06-20T03:56:19.352-07:002010-06-20T03:56:19.352-07:00Option E: cos(α)= 1/9Option E: cos(α)= 1/9Ajithttps://www.blogger.com/profile/00611759721780927573noreply@blogger.com