Two intersecting circles with a triangle connecting them. Show that the three red points are collinear.
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Angle chasing galore
Author: Marshall W. Buck
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Balloon lifting
Three small balloons are in congruent sheds awaiting lift off. The middle shed opens its side flaps. Show that its balloon can inflate, without rolling it, until it is tangent to the sides of the adjacent sheds.
The red and green wedges subtend the same angle, there are two right angles, and the blue sides have the same length. What is the ratio of lengths of the green and red segments?
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Downhill skiing
Choose any point D on the side AB of an isosceles triangle ABC, then extend the side AC beyond C to a point F with |CF| = |DB|. Show that the segment DF is cut exactly in two by the third side BC—that the orange and red trails are equal.
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Ellipsoidal laser
The inside surface of an ellipse is a perfect mirror. There is a pin-hole at an end of the diameter. Show that a light ray emitted from either focus will exit the enclosure via the pin-hole, perhaps after bouncing through the foci several times.
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Corner pursuit
Four corners of a square pursue each other clockwise, and after each unit step mark out the new smaller square. Show that the squares sizes in the sequence decrease in size to 1 in the limit.
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The hexagonal dart board
25 darts land inside a regular-hexagonal dart board with edge length the square root of 3. Show that at least five of the darts land inside the same unit circle.
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Wrapping presents
Six squares are wrapped together with red ribbons. Do they cross at a single point?
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Hidden circle II
What is the locus of orange points for which the purple lines are parallel? The fixed cevians of the triangle BCD are angle bisectors.
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Rolling level
Blue and green circles with tangents. The green tangent line goes through the blue tangent point and the points IJH are collinear. Show that the blue and orange lines are parallel.