A ball is balanced on the edge of a square. The chord DH extends to C, and the points D, H project through the point B to points M, K on the bottom. Show that the midpoint of MK lines up vertically with I, the circle’s leftmost point.
A semicircle and an orange triangle with its orthocentre H. What’s the angle α?
Two squares inscribed in a circle. Prove that the red triangle is equilateral.
Three congruent squares and an extended side. Prove that the three red points are collinear.
The green dome city has beam projectors tangent to the dome. So does the orange dome city. Where the beams from one city meet the beams from the other city, an explosion occurs in the sky. Show that the explosions are concyclic.
A rectangle containing a circle with two tangents. Prove that the red line segments are parallel.
Green and blue tangent circles, with centers A, D. Parallelogram ADCB, with C on the edge of the green. Red semicircle on diameter AC. Dotted circle through C with center B. Show that the green, red, and dotted arcs have a common point.
Four equilateral triangles inside a square. The green triangles are congruent. Prove that the three red vertices are collinear.
Show that the points B, H, I, and G are concentric. (Equivalently, show that angle HGI is a right angle.)
A semicircle and three squares. Prove that D is the incentre of triangle ABC.
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