258 /www/nrich/html/content/99/03/15plus3/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"><br></br> <p>$O$ is the centre of a circle with $A$ and $B$ two points NOT on a diameter. The tangents to $A$ and $B$ intersect at $C$. $CO$ cuts the circle at $D$ and a tangent through $D$ cuts $AC$ and $BC$ at $E$ and $F$.</p> <p>What is the relationship between area of $ADBO$ and the areas of $ABO$ and $ACBO$?</p> <p> </p></mdoxml> <mdoxml version="1.0"><br></br> <table border="0"> <tbody> <tr> <td><mdo:image height="310" width="258" src="mean.gif" alt="diagram"></mdo:image></td> <td> <div>This involves nothing more than areas of right angled triangles, using the symmetry in the diagram, and sines, cos's and tan's.</div> <br></br> <div>Well done M.S. Ezzeri Esa from Cambridge Tutors College, Croydon and thank you for this solution.</div> <br></br> <div>Let radius = $r$; $\angle AOD = \angle BOD = \alpha$</div> <div>Area $ADBO$ = $2 ({1\over 2 }r^2 \sin \alpha) = r^2 \sin \alpha$</div> <div>Area $ABO$ = ${1\over 2}r^2 \sin 2\alpha = r^2 \sin \alpha \cos \alpha$</div> <div>Area $ACBO$ = $2({1\over 2}r^2 \tan \alpha) = r^2 \tan \alpha$</div> <div>(Area $ABO$). (Area $ACBO$) = $r^2 \sin \alpha \cos \alpha\ .\ r^2 \tan \alpha = r^4 \sin^2 \alpha ={\rm (Area ADBO)}^2.$</div> <br></br> <div>The area of $ADBO$ is the geometric mean of the areas of $ABO$ and $ACBO$</div> </td> </tr> </tbody> </table> <br></br> <br></br> <br clear="all"></br> <br></br></mdoxml> <mdoxml version="1.0"><p>Area fomulas and a little trig will help.</p> <p> </p> <p>Here is the diagram if you need some clues to get started:</p> <p><mdo:image src="mean.gif"></mdo:image></p></mdoxml> 2 3 0 0 0 0 1 A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO? Admin Stage 5 - Reviewed 2012 Advanced Algebra Geometric mean Measures and Mensuration Area Transformations and their Properties Symmetry Measures and Mensuration Arcs, sectors and segments Collections epsilons