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
Prove that the internal angle bisectors of a triangle will never be
perpendicular to each other.
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.
The area of triangle $ABC$ is double the area of $AED$.
$X$ is any suitable point on $AD$
$ZX$ is perpendicular to $AC$, and $ZX$ is equal in length to
So $AXZ$ is an isosceles right-angled triangle.
By sweeping an arc centre $A$ from $X$ to $AZ$ at $N$, $AN$ is
made equal to $AX$
$AN$ to $AZ$ is now in the required ratio.
Drawing from $N$ parallel to $ZC$ the point $D$ is reached.
Because $AND$ and $AZC$ are similar triangles, $AD$ and $AC$ are
in the required ratio.
Excellent and simple!