A circular mountain peak has coordinates (0,0,3) and its base has towns C=(-4,0,0), E=( 4,0,0) on opposite sides. Town D=(0,4,0) is halfway between them, around the base. A road goes partially up and down the mountain connecting C and E. Another road connects C and D. Both roads are shortest possible. What is the ratio of the road lengths?
Two similar triangles ABC and EBD. Two line segments meet at point F. Prove that ABFC and EBDF are cyclic quadrilaterals.
A regular pentagon and a circular arc through its red centre. What’s the proportion α : β?
A regular nonagon and four equilateral triangles. The purple and yellow triangles share a vertex. Prove that the yellow triangles also share a vertex.
An equilateral triangle with a circular arc through its red centre and two vertices. If the yellow area equals the green one, what’s the angle α?
The heart is divided into right triangles, each of which contains a blue circle. What is the total circumference of the blue circles in relation to the green perimeter and the lengths of the purple and orange lines in the centre?
A regular nonagon and two equilateral triangles. What’s the angle α?
A regular hexagon with two diagonals and two perpendiculars. What’s red : yellow?
A triangle with three concurring cevians. What’s orange : blue?
The blue quadrilateral contains a purple interior point which is a corner for two right angles. Reflect the purple point about the four sides of the quadrilateral. Show that the four image points are concyclic.