Two nested right triangles with their incircles. Two tangency points are shown. Prove that the red and yellow line segments are parallel.
Tag: circle
The hippodrome shadow
The racer at A is running counter-clockwise around the inside of the outer purple circle, with a “shadow racer” B following behind, related by the three chords that are tangent to the inner orange circle. Does the shadow B ever catch up with A?
Cherries
Two tangent circles and two green tangents. The three tangency points are shown in black. Prove that the blue line segment is a common tangent.
The punctured hexagon
A regular hexagon, a square and an equilateral triangle. Show that the top vertex of the triangle is on the hexagon circumcircle.
Circle triplet
Identical circles have their centers aligned as shown, with the middle circle inscribed in a triangle. Show that the outer circles can be made tangent to the green circumcircle if the red axis is rotated correctly.
Blowing bubbles
Three congruent blue circles with a common intersection point. Prove that the red circle through the other intersections is also congruent.
Colour convergence
A green line connects two points of tangency of a triangle to its incircle. A red line is the angle bisector at a vertex, and an orange line connects the midpoints of two sides. Show that the three lines are concurrent.
In and out
An obtuse triangle, of which α is the obtuse angle, with its incircle, two cevians and two tangents. Find a relation between α and β.
Two quarter-circles fit inside a square, and five circles fit inside the overlapped region, centred. What is the ratio of the (straight line) segment lengths red to blue? (The segments connect to points of tangency.)
Pop you bubble
A square with a semicircle and a circle of equal radius. Their tangency point is shown. Prove that the red triangle is equilateral.