tag:blogger.com,1999:blog-6933544261975483399.post9122960281201923928..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Problem 357. Square, Exterior point, Triangles, AreaAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger4125tag:blogger.com,1999:blog-6933544261975483399.post-1616666213225473222020-11-19T22:21:44.638-08:002020-11-19T22:21:44.638-08:00Draw a Perpendicular from P equal to the length of...Draw a Perpendicular from P equal to the length of side of the square (say h).<br />Denote it as O.Connect AO & DO to form two parallelograms ABPO,DCPO and a triangle AOD congruent to BPC.<br />It can easily seen that S1+S2+S3=S4<br />Hence S1+S3= S4-S2<br />Let the length of Perpendicular from P to BC be h1<br />=> S4-S2 = 0.5(h1.h+h2-h1.h)=h^2/2=S/2Sailendra Thttps://www.blogger.com/profile/12056621729673423024noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-10569187465843809712020-11-10T05:07:12.324-08:002020-11-10T05:07:12.324-08:00See the drawing
Define a = side of square ABCD
De...See the <a href="http://sciences.heptic.fr/2020/11/10/gogeometry-problem-357/" rel="nofollow"><b>drawing</b></a><br /><br />Define a = side of square ABCD<br />Define h1,h2,h3 and h4 the heights respectively of S1, S2, S3 and S4<br />h1+h3=a, h4-h2=a<br />S=[ABCD]=a^2<br />2S1=a.h1, 2S3=a.h3<br />2S1+2S3=a(h1+h3)=a^2<br />2S2=a.h2, 2S4=a.h4<br />2S4-2S2=a(h4-h2)=a^2<br />Therefore <b>S1+S3=S4-S2=S/2</b><br /><br />rv.littlemanhttps://www.blogger.com/profile/15820037721044128612noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-30693629336154113142016-06-18T00:47:34.221-07:002016-06-18T00:47:34.221-07:00Draw EPF//BC, E on AB extended and F on DC extende...Draw EPF//BC, E on AB extended and F on DC extended<br /><br />Let EP = x and EB = y<br /><br />So S1 = xa/2, S3 = (a-x)a/2, <br />S2 = ya/2 and S4 = (a+y)a/2<br /><br />So S1+S3 = S4-S2= a^2/2 = S/2<br /><br />Sumith Peiris<br />Moratuwa<br />Sri LankaSumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-17022780374654718102009-09-22T09:25:12.178-07:002009-09-22T09:25:12.178-07:00Let l be the side of the square ABCD and h1, h2, h...Let l be the side of the square ABCD and h1, h2, h3, h4 be respectively the heights of triangles PAB, PBC, PCD, PDA. We have:<br />S↓1=lh↓1/2, S↓2 = lh↓2/2, S↓3=l h↓3/2, S↓4=l h↓4/2. Therefore:<br /><br />S↓1+ S↓3=l(h↓1+ h↓3)/2= l↑2./2,<br />S↓4- S↓2=l(h↓4- h↓2)/2= l↑2./2.<br />QED, Ianuarius.Anonymousnoreply@blogger.com