A regular 7-gon inscribed in a circle (ABCDEFG), with some other point H on the circle, in the arc CD. Show that the length sum of the blue chords equals the length sum of the green chords.
Tag: length
The gondola lift
A tangent and its normal in an arbitrary point on an ellipse with eccentricity e. The focal points are also shown. What is the minimum value of blue : red in terms of e?
The grassy knoll
A triangle with three cevians that are concurrent in an arbitrary interior point. What is the maximum value of bdf/ace?
The tipi entrance
A semicircle with three tangents. Prove that purple : green = red : yellow.
Baby block II
A square containing another square with an extended side. What is tan(α) in terms of the red and green segment lengths?
Flying purple people eater
What is the area of the purple people eater, in terms of the lengths AD and BC?
Mountain roads
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?
Mouth to mouth
What is the ratio of the blue and yellow lines?
Equal rights
Two squares sharing a vertex. Two vertices are connected by a line segment. What is yellow : green?
Holes in my heart
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?