{"id":418,"date":"2019-12-17T18:29:44","date_gmt":"2019-12-17T18:29:44","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=418"},"modified":"2020-10-15T18:30:42","modified_gmt":"2020-10-15T18:30:42","slug":"circles-in-a-square-part-10","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=418","title":{"rendered":"Circles in a Square – Part 10"},"content":{"rendered":"\n

In this 10th post of this series, consider the following figure with two co-tangent semi-circles drawn, centered at point $E$ on\u00a0$\\overline{AB}$ and $F$ on $\\overline{BC}$. A full circle centered at point G,\u00a0tangent\u00a0with both\u00a0semi-circles,\u00a0line $\\overline{AD}$ and $\\overline{CD}$ in a unit\u00a0square\u00a0$ABCD$. Find the radii of the\u00a0semi-circles\u00a0and the\u00a0full circle?\u00a0Click here to show the solution.<\/a><\/p>\n\n\n\n

\"\"<\/figure><\/div>\n\n\n\n

Assume the radius of the two congruent semi-circle is $r$, the radius of the full circle is $s$. First, we need to find the value of $r$, as as the semi-circles are drawn first in the set up. Let’s draw line $\\overline{EF}$, connecting the centers of the semi-circles, as shown below:<\/p>\n

\"\"<\/figure><\/div>\nObviously, $\\triangle{BEF}$ is a right triangle. We have $$\\overline{EF}^2=\\overline{BE}^2+\\overline{BF}^2$$ Since $\\overline{AE}=\\overline{CF}=r$, $\\overline{BE}=\\overline{BF}=\\overline{AB}-\\overline{AE}=1-r$, we have: $$(2r)^2=(1-r)^2+(1-r)^2$$\nThe solution of the above equation, we have $r=\\pm\\sqrt{2}-1$. Ignoring the negative $r$ value, we have the radius of the semi-circles as\n\n$$r=\\sqrt{2}-1$$\n\nNext, we consider the full circle center at point $G$. Connect the center of one semi-circle and the center point of the full circle, we have $\\overline{EG}$ intersecting the semi and full circles at point $I$. We have\n$$\\overline{EG}=\\overline{EI}+\\overline{GI}=\\sqrt{2}-1+s$$\nDraw line $\\overline{KL}$ passing thru $G$ and parallel to $\\overline{CD}$, and line $\\overline{EM}$ perpendicular to $\\overline{CD}$ and intersecting with line $\\overline{KL}$ at point $N$. As $\\triangle{EGN}$ is a right triangle, we have $$\\overline{EG}^2=\\overline{EN}^2+\\overline{GN}^2$$\nSince $$\\overline{EN}=\\overline{EM}-\\overline{NM}=1-s$$\n$$\\overline{GN}=\\overline{KN}-\\overline{GK}=\\overline{AE}-s=\\sqrt{2}-1-s$$\nTherefore\n$$(\\sqrt{2}\u22121+s)^2=(1\u2212s)^2+(\\sqrt{2}\u22121\u2212s)^2$$\nSolve the above equation, we have $$s=-1+2\\sqrt{2}\\pm2\\sqrt{2-\\sqrt{2}}$$\nIgnoring the $s$ value greater than 1, we have the radius of the full circle as $$s=-1+2\\sqrt{2}-\\sqrt{2-\\sqrt{2}} \\approx 0.2976933952$$\n<\/p>\n<\/div>\n\n\n\n

To be continued…<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"

In this 10th post of this series, consider the following figure with two co-tangent semi-circles drawn, centered at point $E$ on\u00a0$\\overline{AB}$ and $F$ on $\\overline{BC}$. A full circle centered at point G,\u00a0tangent\u00a0with both\u00a0semi-circles,\u00a0line $\\overline{AD}$ and $\\overline{CD}$ in a unit\u00a0square\u00a0$ABCD$. Find … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[12,9],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/418"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=418"}],"version-history":[{"count":16,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/418\/revisions"}],"predecessor-version":[{"id":436,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/418\/revisions\/436"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=418"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=418"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=418"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}