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Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST and PU are perpendicular to AB produced. Show that ST + PU = AB

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A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

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Angle Trisection

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Encircling

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Congratulations James Page of Hethersett High School, Norwich on your solution to this question. Again there are many possible methods of showing that the radius of the circle is equal to the side of the square.



The blue line $OD$ is a diameter of the circle and a line of symmetry.

$\angle EDC = 150^{\circ}$ and $\triangle ECD$ isosceles so $\angle CED = \angle CDE =15^{\circ}$.

Now $\angle OED = 30^{\circ}$ so $\angle OEC = \angle OED - \angle CED = 15^{\circ}$ and hence the red line $EC$ is a line of symmetry for the quadrilateral $OEDC$ proving that the radius of the circle, $OC$ is equal in length to the side of the square.