Problems

A maths teacher draws a number of circles on a piece of paper. When she shows this piece of paper to the young mathematician, he claims he can see only five circles. The maths teacher agrees. But when she shows the same piece of paper to another young mathematician, he says that there are exactly eight circles. The teacher confirms that this answer is also correct. How is that possible and how many circles did she originally draw on that piece of paper?

What is the ratio between the red and blue area? All shapes are semicircles.

The marked pink segment (tangent to the inner circle) has length \(1\). Find the area of the green annulus.

Point \(A\) is the centre of a circle and points \(B,C,D\) lie on that circle. The segment \(BD\) is a diameter of the circle. Show that \(\angle CAD = 2 \angle CBD\). This is a special case of the inscribed angle theorem.

Point \(A\) is the centre of a circle and points \(B,C,D\) lie on that circle. Show that \(\angle CAD = 2 \angle CBD\). This statement is known as the inscribed angle theorem and is used widely in Euclidean geometry.

Let \(BCDE\) be a quadrilateral inscribed in a circle with centre \(A\). Show that angles \(\angle CDE\) and \(\angle CBE\) are equal. Also show that angles \(\angle BCD\) and \(\angle BED\) are equal. This says that all angles at the circumference subtended by the same arc are equal.

Let \(BCDE\) be an inscribed quadrilateral. Show that \(\angle BCD + \angle BED = 180^{\circ}\).

The points \(B\),\(C\),\(D\),\(E\),\(F\) and \(G\) lie on a circle with centre \(A\). The angles \(\angle CBD\) and \(\angle EFG\) are equal. Prove that the segments \(CD\) and \(EG\) have equal lengths.

On the diagram below find the value of the angles \(\angle CFD\) and \(\angle CGD\) in terms of angles \(\angle CBD\) and \(\angle BDE\).

Point \(A\) is the centre of a circle. Points \(B,C,D,E\) lie on the circumference of this circle. Lines \(BC\) and \(DE\) cross at \(F\). We label the angles \(\angle BAD =\delta\) and \(\angle CAE = \gamma\). Express the angle \(DFB\) in terms of \(\gamma\) and \(\delta\).