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Difference between revisions of "2004 IMO Problems/Problem 5"

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Let <math>K</math>  be the intersection of <math>AC</math> and <math>BE</math>, let <math>L</math> be the intersection of <math>AC</math> and <math>DF</math>,
 
Let <math>K</math>  be the intersection of <math>AC</math> and <math>BE</math>, let <math>L</math> be the intersection of <math>AC</math> and <math>DF</math>,
 +
 +
<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px"
 +
        width="1000px" height="1200px" viewBox="0 0 1000 1200" enable-background="new 0 0 1000 1200" xml:space="preserve">
 +
<text transform="matrix(1 0 0 1 219 680.5)" font-family="'MyriadPro-Regular'" font-size="12">A</text>
 +
<text transform="matrix(1 0 0 1 258.5 666)" font-family="'MyriadPro-Regular'" font-size="12">K</text>
 +
<text transform="matrix(1 0 0 1 296.4033 650.5)" font-family="'MyriadPro-Regular'" font-size="12">L</text>
 +
<text transform="matrix(1 0 0 1 269.1636 696.5)" font-family="'MyriadPro-Regular'" font-size="12">B</text>
 +
<text transform="matrix(1 0 0 1 345.0273 617.2539)" font-family="'MyriadPro-Regular'" font-size="12">C</text>
 +
<text transform="matrix(1 0 0 1 203.353 574.9707)" font-family="'MyriadPro-Regular'" font-size="12">D</text>
 +
<text transform="matrix(1 0 0 1 276.5273 617.5)" font-family="'MyriadPro-Regular'" font-size="12">P</text>
 +
<text transform="matrix(1 0 0 1 275 540.5)" font-family="'MyriadPro-Regular'" font-size="12">E</text>
 +
<text transform="matrix(1 0 0 1 325.5 666)" font-family="'MyriadPro-Regular'" font-size="12">F</text>
 +
<circle fill="none" stroke="#000000" stroke-miterlimit="10" cx="269.164" cy="614.164" r="69.164"/>
 +
<line fill="none" stroke="#000000" stroke-miterlimit="10" x1="230.387" y1="671.441" x2="338.327" y2="620.34"/>
 +
<line fill="none" stroke="#000000" stroke-miterlimit="10" x1="210.979" y1="576.758" x2="278.062" y2="545"/>
 +
<line fill="none" stroke="#000000" stroke-miterlimit="10" x1="269.164" y1="683.327" x2="323.643" y2="656.776"/>
 +
<line fill="none" stroke="#000000" stroke-miterlimit="10" x1="269.164" y1="683.327" x2="278.062" y2="545"/>
 +
<line fill="none" stroke="#000000" stroke-miterlimit="10" x1="210.979" y1="576.758" x2="323.643" y2="656.776"/>
 +
<line fill="none" stroke="#000000" stroke-miterlimit="10" x1="230.387" y1="671.441" x2="210.979" y2="576.758"/>
 +
<line fill="none" stroke="#000000" stroke-miterlimit="10" x1="210.979" y1="576.758" x2="338.327" y2="620.34"/>
 +
<line fill="none" stroke="#000000" stroke-miterlimit="10" x1="230.387" y1="671.441" x2="269.164" y2="683.327"/>
 +
<line fill="none" stroke="#ED1C24" stroke-miterlimit="10" stroke-dasharray="2,2" x1="338.327" y1="620.34" x2="278.062" y2="545"/>
 +
<line fill="none" stroke="#ED1C24" stroke-miterlimit="10" stroke-dasharray="2,2" x1="323.643" y1="656.776" x2="338.327" y2="620.34"/>
 +
<line fill="none" stroke="#ED1C24" stroke-miterlimit="10" x1="230.387" y1="671.441" x2="273.177" y2="620.934"/>
 +
<line fill="none" stroke="#ED1C24" stroke-miterlimit="10" x1="273.177" y1="620.934" x2="338.327" y2="620.34"/>
 +
<g>
 +
        <g>
 +
                <path d="M271.5,654c3.224,0,3.224-5,0-5S268.276,654,271.5,654L271.5,654z"/>
 +
        </g>
 +
</g>
 +
<g>
 +
        <g>
 +
                <path d="M299.5,641.5c3.224,0,3.224-5,0-5S296.276,641.5,299.5,641.5L299.5,641.5z"/>
 +
        </g>
 +
</g>
 +
<line fill="none" stroke="#000000" stroke-miterlimit="10" x1="269.164" y1="683.327" x2="338.327" y2="620.34"/>
 +
<line fill="none" stroke="#000000" stroke-miterlimit="10" x1="210.979" y1="576.758" x2="269.164" y2="683.327"/>
 +
</svg>
  
 
==See Also==
 
==See Also==
  
 
{{IMO box|year=2004|num-b=4|num-a=6}}
 
{{IMO box|year=2004|num-b=4|num-a=6}}

Revision as of 15:32, 8 February 2024

Problem

In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies \[\angle PBC = \angle DBA \text{ and } \angle PDC = \angle BDA.\]

Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP = CP.$

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

Let $K$ be the intersection of $AC$ and $BE$, let $L$ be the intersection of $AC$ and $DF$,

<svg version="1.1" id="Layer_1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" 

        width="1000px" height="1200px" viewBox="0 0 1000 1200" enable-background="new 0 0 1000 1200" xml:space="preserve">

<text transform="matrix(1 0 0 1 219 680.5)" font-family="'MyriadPro-Regular'" font-size="12">A</text> <text transform="matrix(1 0 0 1 258.5 666)" font-family="'MyriadPro-Regular'" font-size="12">K</text> <text transform="matrix(1 0 0 1 296.4033 650.5)" font-family="'MyriadPro-Regular'" font-size="12">L</text> <text transform="matrix(1 0 0 1 269.1636 696.5)" font-family="'MyriadPro-Regular'" font-size="12">B</text> <text transform="matrix(1 0 0 1 345.0273 617.2539)" font-family="'MyriadPro-Regular'" font-size="12">C</text> <text transform="matrix(1 0 0 1 203.353 574.9707)" font-family="'MyriadPro-Regular'" font-size="12">D</text> <text transform="matrix(1 0 0 1 276.5273 617.5)" font-family="'MyriadPro-Regular'" font-size="12">P</text> <text transform="matrix(1 0 0 1 275 540.5)" font-family="'MyriadPro-Regular'" font-size="12">E</text> <text transform="matrix(1 0 0 1 325.5 666)" font-family="'MyriadPro-Regular'" font-size="12">F</text> <circle fill="none" stroke="#000000" stroke-miterlimit="10" cx="269.164" cy="614.164" r="69.164"/> <line fill="none" stroke="#000000" stroke-miterlimit="10" x1="230.387" y1="671.441" x2="338.327" y2="620.34"/> <line fill="none" stroke="#000000" stroke-miterlimit="10" x1="210.979" y1="576.758" x2="278.062" y2="545"/> <line fill="none" stroke="#000000" stroke-miterlimit="10" x1="269.164" y1="683.327" x2="323.643" y2="656.776"/> <line fill="none" stroke="#000000" stroke-miterlimit="10" x1="269.164" y1="683.327" x2="278.062" y2="545"/> <line fill="none" stroke="#000000" stroke-miterlimit="10" x1="210.979" y1="576.758" x2="323.643" y2="656.776"/> <line fill="none" stroke="#000000" stroke-miterlimit="10" x1="230.387" y1="671.441" x2="210.979" y2="576.758"/> <line fill="none" stroke="#000000" stroke-miterlimit="10" x1="210.979" y1="576.758" x2="338.327" y2="620.34"/> <line fill="none" stroke="#000000" stroke-miterlimit="10" x1="230.387" y1="671.441" x2="269.164" y2="683.327"/> <line fill="none" stroke="#ED1C24" stroke-miterlimit="10" stroke-dasharray="2,2" x1="338.327" y1="620.34" x2="278.062" y2="545"/> <line fill="none" stroke="#ED1C24" stroke-miterlimit="10" stroke-dasharray="2,2" x1="323.643" y1="656.776" x2="338.327" y2="620.34"/> <line fill="none" stroke="#ED1C24" stroke-miterlimit="10" x1="230.387" y1="671.441" x2="273.177" y2="620.934"/> <line fill="none" stroke="#ED1C24" stroke-miterlimit="10" x1="273.177" y1="620.934" x2="338.327" y2="620.34"/> <g> 

       <g>
               <path d="M271.5,654c3.224,0,3.224-5,0-5S268.276,654,271.5,654L271.5,654z"/>
       </g>

</g> <g> 

       <g>
               <path d="M299.5,641.5c3.224,0,3.224-5,0-5S296.276,641.5,299.5,641.5L299.5,641.5z"/>
       </g>

</g> <line fill="none" stroke="#000000" stroke-miterlimit="10" x1="269.164" y1="683.327" x2="338.327" y2="620.34"/> <line fill="none" stroke="#000000" stroke-miterlimit="10" x1="210.979" y1="576.758" x2="269.164" y2="683.327"/> </svg>

See Also

2004 IMO (Problems) • Resources
Preceded by
Problem 4 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 6
All IMO Problems and Solutions
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