A circle touches the lines OA, OB and AB where OA and OB are
perpendicular. Show that the diameter of the circle is equal to the
perimeter of the triangle
P is a point on the circumference of a circle radius r which rolls,
without slipping, inside a circle of radius 2r. What is the locus
I keep three circular medallions in a rectangular box in which they
just fit with each one touching the other two. The smallest one has
radius 4 cm and touches one side of the box, the middle sized one
has radius 9 cm and touches two sides of the box and the largest
one touches three sides of the box. What is the radius of the
Let T be the centre of the semicircle with diameter QR and let OT produced meet the circumference of the larger semicircle at U .
By symmetry, we note that OT is perpendicular to QR . As TR = TO = TQ (radii of the same semicircle), triangles ORT and OQT are both isosceles, right-angled triangles. So QOR is a right angle.
By Pythagoras' Theorem: QR ² = OQ ² + OR ² = 2² + 2² = 8. So $QR = \sqrt8$ = $2\sqrt2$ and the radius of semicircle QOR is $\sqrt2$.
This problem is taken from the UKMT Mathematical Challenges.