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
Three circular medallions fit in a rectangular box. Can you find the radius of the largest one?
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.