Fixed rotor (triangle) CDE inscribed in purple circle. Moving point F on the circle. Define for the point F a new rotor HJL on the circle, such that FH, FJ, FL are perpendicular to CD, CE, and DE respectively. Show that the green triangles HJL do not change shape as F moves.

# Author: Marshall W. Buck

## Square construction

Given the blue square AEFI and a point B inside, show that the intersection H of the red and purple semicircles will be a corner of a square containing B on one side and A on another, and sharing the corner I with the blue square.

## The whirligig

The green blades form a bowtie that is symmetric about the vertical orange line g. Show that the blue horizontal line f, the red circle (ADE), and the purple circle (AOC) are concurrent. (Assume that COB and AOD are colinear.)

Blue and orange concentric circles intersect a vertical line in green dots. Show that the dots are equally spaced, given that the red intervals shown are of equal length. (The largest circle has radius 2u.)

## Equilateral linkage

Given 3 points A, B, C, define G so CG=CB and angle GCB is 120 degrees. Define M as the midpoint of side CH of the parallelogram CGAH. The green equilateral has side CM and the purple equilateral has side MB. Is AJK also equilateral?

## The line up

Two semicircles share a common diameter HB. Point A in the upper half projects to E along the diameter, and to G along the line segment BC. F is the midpoint of AC. Show that FEG are collinear.

## Circle a ranch brand

Connect 4 (blue) points B, C, D, E on a circle to some of the (orange) midpoints of non-adjacent arcs. (For example, C connects to J, E connects to L.) Intersect CJ with EL to get F, and intersect DK with BL to get G. Show that FG || JK.

## Circle crossings

The segment CD is tangent to the green circle c, ending on the orange circle d, which also contains the foot of the perpendicular line from C to the diameter of c which extends to D. What is the angle between the two circles?

## The dunce cap

The orthocentre H, circumcentre O, the incentre I, and points B and C all lie on the top of the dunce’s spherical head. What is the angle α?

## Hidden parallelogram

Blue circle (ACDGB) with CEG || AB. Pink circle (EDG) intersects AB at I. Extend DI to intersect blue circle at J, then JC intersects AB at M. Show that IM = CE.