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문제

삼각형 안에 세 개의 원을 각각의 원이 다른 두 원과 삼각형의 두 변에 접하게 그릴 수 있다. 이러한 원을 말파티 원(Malfatti Circles)이라고 한다. 많은 수학자들이 말파티 원을 두 세기 이상 연구하였다. 임의의 삼각형에 대해서 말파티 원은 항상 존재하고 유일하다는 것이 증명되어 있다.

삼각형의 꼭지점이 (20, 80), (-40, -20), (120,-20)일 때, 삼각형의 말파티 원은 다음과 같다.

  • 원의 중심이 (24.281677, 45.219486), 반지름이 21.565935
  • 원의 중심이 (3.110950, 4.409005), 반지름이 24.409005
  • 원의 중심이 (54.556724, 7.107493), 반지름이 27.107493

또, 삼각형의 꼭지점이 (20, -20), (120, -20), (-40, 80)일 때, 삼각형의 말파티 원은 다음과 같다.

  • 원의 중심이 (25.629089, -10.057956), 반지름이 9.942044
  • 원의 중심이 (53.225883, -0.849435), 반지름이 19.150565
  • 원의 중심이 19.701191, 19.203466), 반지름이 19.913790

삼각형이 주어졌을 때, 말파티 원을 구하는 프로그램을 작성하시오.

입력

입력은 여러 개의 테스트 케이스로 이루어져 있다. 각 테스트 케이스는 여섯 정수 x1, y1, x2, y2, x3, y3이 공백으로 구분되어져 있다. 이 좌표는 삼각형의 각 꼭지점이고 (x1, y1), (x2, y2), (x3, y3)이다. 입력으로 주어지는 삼각형의 꼭지점은 반시계방향 순서이고 다음과 같은 조건을 만족한다.

  1. 모든 꼭지점의 좌표는 -1000보다 크고 1000보다 작다.
  2. 반지름이 0.1보다 작은 말파티 원은 없다.

입력의 마지막 줄에는 0이 여섯 개 주어진다.

출력

각 테스트 케이스에 대해서 말파티 원의 반지름 r1, r2, r3를 소수점 여섯째 자리까지 출력한다. ri는 꼭지점 (xi, yi)와 가장 가까운 원의 반지름이다.

출력하는 반지름의 오차는 0.0001보다 크면 안된다.

예제 입력 1

20 80 -40 -20 120 -20
20 -20 120 -20 -40 80
0 0 1 0 0 1
0 0 999 1 -999 1
897 -916 847 -972 890 -925
999 999 -999 -998 -998 -999
-999 -999 999 -999 0 731
-999 -999 999 -464 -464 999
979 -436 -955 -337 157 -439
0 0 0 0 0 0

예제 출력 1

21.565935 24.409005 27.107493
9.942044 19.150565 19.913790
0.148847 0.207107 0.207107
0.125125 0.499750 0.499750
0.373458 0.383897 0.100456
0.706768 0.353509 0.353509
365.638023 365.638023 365.601038
378.524085 378.605339 378.605339
21.895803 22.052921 5.895714

힌트

[{"problem_id":"3881","problem_lang":"0","title":"\ub9d0\ud30c\ud2f0 \uc6d0","description":"<p>\uc0bc\uac01\ud615 \uc548\uc5d0 \uc138 \uac1c\uc758 \uc6d0\uc744 \uac01\uac01\uc758 \uc6d0\uc774 \ub2e4\ub978 \ub450 \uc6d0\uacfc \uc0bc\uac01\ud615\uc758 \ub450 \ubcc0\uc5d0 \uc811\ud558\uac8c \uadf8\ub9b4 \uc218 \uc788\ub2e4. \uc774\ub7ec\ud55c \uc6d0\uc744 \ub9d0\ud30c\ud2f0 \uc6d0(Malfatti Circles)\uc774\ub77c\uace0 \ud55c\ub2e4. \ub9ce\uc740 \uc218\ud559\uc790\ub4e4\uc774 \ub9d0\ud30c\ud2f0 \uc6d0\uc744 \ub450 \uc138\uae30 \uc774\uc0c1 \uc5f0\uad6c\ud558\uc600\ub2e4. \uc784\uc758\uc758 \uc0bc\uac01\ud615\uc5d0 \ub300\ud574\uc11c \ub9d0\ud30c\ud2f0 \uc6d0\uc740 \ud56d\uc0c1 \uc874\uc7ac\ud558\uace0 \uc720\uc77c\ud558\ub2e4\ub294 \uac83\uc774 \uc99d\uba85\ub418\uc5b4 \uc788\ub2e4.<\/p>\r\n\r\n<p>\uc0bc\uac01\ud615\uc758 \uaf2d\uc9c0\uc810\uc774 (20, 80), (-40, -20), (120,-20)\uc77c \ub54c, \uc0bc\uac01\ud615\uc758 \ub9d0\ud30c\ud2f0 \uc6d0\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\uc6d0\uc758 \uc911\uc2ec\uc774 (24.281677, 45.219486), \ubc18\uc9c0\ub984\uc774 21.565935<\/li>\r\n\t<li>\uc6d0\uc758 \uc911\uc2ec\uc774 (3.110950, 4.409005), \ubc18\uc9c0\ub984\uc774 24.409005<\/li>\r\n\t<li>\uc6d0\uc758 \uc911\uc2ec\uc774 (54.556724, 7.107493), \ubc18\uc9c0\ub984\uc774 27.107493<\/li>\r\n<\/ul>\r\n\r\n<p>\ub610, \uc0bc\uac01\ud615\uc758 \uaf2d\uc9c0\uc810\uc774 (20, -20), (120, -20), (-40, 80)\uc77c \ub54c, \uc0bc\uac01\ud615\uc758 \ub9d0\ud30c\ud2f0 \uc6d0\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\r\n\r\n<ul>\r\n\t<li>\uc6d0\uc758 \uc911\uc2ec\uc774 (25.629089, -10.057956), \ubc18\uc9c0\ub984\uc774 9.942044<\/li>\r\n\t<li>\uc6d0\uc758 \uc911\uc2ec\uc774 (53.225883, -0.849435), \ubc18\uc9c0\ub984\uc774 19.150565<\/li>\r\n\t<li>\uc6d0\uc758 \uc911\uc2ec\uc774 19.701191, 19.203466), \ubc18\uc9c0\ub984\uc774 19.913790<\/li>\r\n<\/ul>\r\n\r\n<p><img alt=\"\" src=\"\/upload\/images\/malfatti.png\" style=\"height:173px; width:615px\" \/><\/p>\r\n\r\n<p>\uc0bc\uac01\ud615\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c, \ub9d0\ud30c\ud2f0 \uc6d0\uc744 \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud558\uc2dc\uc624.<\/p>\r\n","input":"<p>\uc785\ub825\uc740 \uc5ec\ub7ec \uac1c\uc758 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub85c \uc774\ub8e8\uc5b4\uc838 \uc788\ub2e4. \uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\ub294 \uc5ec\uc12f \uc815\uc218 x1, y1, x2, y2, x3, y3\uc774 \uacf5\ubc31\uc73c\ub85c \uad6c\ubd84\ub418\uc5b4\uc838 \uc788\ub2e4. \uc774 \uc88c\ud45c\ub294 \uc0bc\uac01\ud615\uc758 \uac01 \uaf2d\uc9c0\uc810\uc774\uace0 (x1, y1), (x2, y2), (x3, y3)\uc774\ub2e4. \uc785\ub825\uc73c\ub85c \uc8fc\uc5b4\uc9c0\ub294 \uc0bc\uac01\ud615\uc758 \uaf2d\uc9c0\uc810\uc740 \ubc18\uc2dc\uacc4\ubc29\ud5a5 \uc21c\uc11c\uc774\uace0 \ub2e4\uc74c\uacfc \uac19\uc740 \uc870\uac74\uc744 \ub9cc\uc871\ud55c\ub2e4.<\/p>\r\n\r\n<ol>\r\n\t<li>\ubaa8\ub4e0 \uaf2d\uc9c0\uc810\uc758 \uc88c\ud45c\ub294 -1000\ubcf4\ub2e4 \ud06c\uace0 1000\ubcf4\ub2e4 \uc791\ub2e4.<\/li>\r\n\t<li>\ubc18\uc9c0\ub984\uc774 0.1\ubcf4\ub2e4 \uc791\uc740 \ub9d0\ud30c\ud2f0 \uc6d0\uc740 \uc5c6\ub2e4.<\/li>\r\n<\/ol>\r\n\r\n<p>\uc785\ub825\uc758 \ub9c8\uc9c0\ub9c9 \uc904\uc5d0\ub294 0\uc774 \uc5ec\uc12f \uac1c \uc8fc\uc5b4\uc9c4\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8 \ucf00\uc774\uc2a4\uc5d0 \ub300\ud574\uc11c \ub9d0\ud30c\ud2f0 \uc6d0\uc758 \ubc18\uc9c0\ub984 r1, r2, r3\ub97c \uc18c\uc218\uc810 \uc5ec\uc12f\uc9f8 \uc790\ub9ac\uae4c\uc9c0 \ucd9c\ub825\ud55c\ub2e4. ri\ub294 \uaf2d\uc9c0\uc810 (xi, yi)\uc640 \uac00\uc7a5 \uac00\uae4c\uc6b4 \uc6d0\uc758 \ubc18\uc9c0\ub984\uc774\ub2e4.<\/p>\r\n\r\n<p>\ucd9c\ub825\ud558\ub294 \ubc18\uc9c0\ub984\uc758 \uc624\ucc28\ub294 0.0001\ubcf4\ub2e4 \ud06c\uba74 \uc548\ub41c\ub2e4.<\/p>\r\n","hint":"","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"3881","problem_lang":"1","title":"Malfatti Circles","description":"<p>The configuration of three circles packed inside a triangle such that each circle is tangent to the other two circles and to two of the edges of the triangle has been studied by many mathematicians for more than two centuries. Existence and uniqueness of such circles for an arbitrary triangle are easy to prove. Many methods of numerical calculation or geometric construction of such circles from an arbitrarily given triangle have been discovered. Today, such circles are called the Malfatti circles.<\/p>\r\n\r\n<p>Figure 7 illustrates an example. The Malfatti circles of the triangle with the vertices (20, 80), (-40, -20) and (120, -20) are approximately<\/p>\r\n\r\n<ul>\r\n\t<li>the circle with the center (24.281677, 45.219486) and the radius 21.565935,<\/li>\r\n\t<li>the circle with the center (3.110950, 4.409005) and the radius 24.409005, and<\/li>\r\n\t<li>the circle with the center (54.556724, 7.107493) and the radius 27.107493<\/li>\r\n<\/ul>\r\n\r\n<p>Figure 8 illustrates another example. The Malfatti circles of the triangle with the vertices (20, -20), (120, -20) and (-40, 80) are approximately<\/p>\r\n\r\n<ul>\r\n\t<li>the circle with the center (25.629089, -10.057956) and the radius 9.942044<\/li>\r\n\t<li>the circle with the center (53.225883, -0.849435) and the radius 19.150565, and<\/li>\r\n\t<li>the circle with the center (19.701191, 19.203466) and the radius 19.913790.<\/li>\r\n<\/ul>\r\n\r\n<p>Your mission is to write a program to calculate the radii of the Malfatti circles of the given triangles.<\/p>\r\n","input":"<p>The input is a sequence of datasets. A dataset is a line containing six integers x1, y1, x2, y2, x3 and y3 in this order, separated by a space. The coordinates of the vertices of the given triangle are (x1, y1), (x2, y2) and (x3, y3), respectively. You can assume that the vertices form a triangle counterclockwise. You can also assume that the following two conditions hold.<\/p>\r\n\r\n<ul>\r\n\t<li>All of the coordinate values are greater than -1000 and less than 1000.<\/li>\r\n\t<li>None of the Malfatti circles of the triangle has a radius less than 0.1.<\/li>\r\n<\/ul>\r\n\r\n<p>The end of the input is indicated by a line containing six zeros separated by a space<\/p>\r\n","output":"<p>For each input dataset, three decimal fractions r1, r2 and r3 should be printed in a line in this order separated by a space. The radii of the Malfatti circles nearest to the vertices with the coordinates (x1, y1), (x2, y2) and (x3, y3) should be r1, r2 and r3, respectively.<\/p>\r\n\r\n<p>None of the output values may have an error greater than 0.0001. No extra character should appear in the output.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]