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Fitting In

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

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No Right Angle Here

Prove that the internal angle bisectors of a triangle will never be perpendicular to each other.

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Lens Angle

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.


Stage: 4 Short Challenge Level: Challenge Level:1

$\angle BAC = 90^{\circ}$

$A$, $B$ and $C$ are all equidistant from $D$ and therefore lie on a circle whose centre is $D$. $BC$ is a diameter of the circle and $\angle BAC$ is therefore the angle subtended by a diameter at a point on the circumference (the angle in the semicircle).

Alternatively, suppose $\angle ACD = x$. Then, $\angle BAC = x$ also, as $DAC$ is isosceles. This means $\angle CDA = 180 - 2x$ by angles in a triangle, so $\angle ABC = 2x$. Then, as $ADB$ is also isosceles, $\angle BAD = \angle DBA = \frac{1}{2}(180-2x) = 90-x$. Therefore, $\angle BAC = \angle BAD + \angle DAC = 90-x+x=90^\circ$

This problem is taken from the UKMT Mathematical Challenges.