Four similar triangles share one circle as incircle or excircle. Show that orange dotted lines must be concurrent.
Category: Intermediate
Interlocked triangles
Do the blue, green, and orange triangles have the same shape?
Six four
A regular hexagon and an inscribed tilted rectangle. What is the angle α?
Ball of yarn
A cyclic hexagon has concurrent cross diagonals. What is the ratio of the product of the orange sides compared to the product of the purple sides. (ace:bdf).
Sunrise over green mountain
An acute triangle mountain ABC has altitudes BE and CF. The dotted tangent lines to the sunny circumcircle (AEF) at E and F intersect at a point M. Show that M is on the mountain’s base BC.
The silly walk
A circle and several triangles. Prove that the green triangle is isosceles.
The sun hat
Two triangles share a circumcircle and vertex, with one edge of the orange triangle containing the feet of two of the altitudes of the blue triangle. Show that the orange triangle is isosceles.
Uneasy the head
The green zigzag crown segments would extend through either B or C. Show that the arcs along the top are equally spaced.
Giant’s shoulder II
Three equilateral triangles share three vertices. What is blue : red?
Falling in
Start with an acute triangle and form a new triangle from the points of tangency of its inscribed circle. Continue this process to make make the triangle with blue vertices. What is the maximum possible angle at a blue vertex?