EHF is tangent to the purple circle at E, and EN is tangent to the red circle at N. A and C are the circle centers. A blue rectangle has three corners HEC and the point A on one side. What is the ratio of areas, green square to orange quadrilateral?

# Author: Marshall W. Buck

## Locus pocus II

What is the locus of points P for which perpendicular lengths (red=l, green=n, purple=m) to the sides of an isosceles triangle satisfy the condition red*green = purple^{2}?

## The center falls apart

In a right triangle ACB show that the incircle touches sides AC and CB in points that are vertically lined up with the centers of the incircles for ADC and BDC. Furthermore, show that the red and green distances are equal.

## Conway circular area

A triangle with sides a, b, c, when extended to whiskers of opposite side length, forms the “Conway Circle”. What is the area of the circle in terms of the expressions a+b+c, ab+bc+ca, abc of the side lengths a, b, c.

## Drilling site

A blue flagpole is erected near a drilling site. Lines are drilled perpendicular to the flagpole guy wires and meet at the bottom of the orange drilling pipe. What is the ratio of distances green to red?

## Rolling along

Tangents to the blue circle (ABC) intersect at a point K, and the line AK intersects (ABC) at H. D is the midpoint of BC. Show that the green circle (DHB) is tangent to the line AB.

## Cathedral on the hills

A cathedral is erected on two hills, the side circular arcs whose centers are the hill ends, and so that the right (and left) side arcs are orthogonal. Show that the tip of the spire is directly above where the hills meet.

## Tangent latitudes

Tangent lines QC and EC meet at C. A point D on QC has DC=1 and QD=2. The line ED intersects the circle at G, and the line HGI is parallel to QDC. What is HG/GI?

## Touch the sides

What is the area of a green quadrilateral that fits inside a quarter circle, and has perpendicular diagonals?

## Flying saucer II

If X is the midpoint of the segment connecting the centers of the upper and lower arcs, and JK is perpendicular to XH, then show that H is the midpoint of JK.