Categories
Advanced

The freight elevator

Four points A, B, C and D form a quadrilateral. Prove that it is a rectangle if and only if for an arbitrary point P we have PA2+PC2=PB2+PD2.

Categories
Intermediate

The augmented chord

A circle with four chords. What is the length of line segment x in terms of lengths a and b?

Categories
Advanced

Fun with flags

A triangle with two cevians. What is d/c-b/a?

Categories
Advanced

Mind the gap

A regular hexagon with an equilateral triangle sharing a vertex and having a vertex somewhere along the opposite side. What is the maximum length of line segment g?

Categories
Intermediate

Pulling the strings

Two rods of length x and y are connected by two strings with a knot exactly halfway. The rods are placed parallel both with uncrossed and crossed strings. The distance between the knots is measured as a and b respectively. Express x and y in terms of a and b.

Categories
Beginner

The tornado

A quadrilateral covers exactly half of the area of a rectangle. Find the length x in terms of a, b and c.

Categories
Intermediate

The steam iron

The midpoints of the hypotenuse of a right triangle and a line segment are connected. What is the length x?

Categories
Intermediate

The major chord

A circle with four chords. What is the segment length x in terms of lengths a, b and c?

Categories
Intermediate

The hawfinch

An equilateral triangle with a circle tangent to two sides. The vertices are the tangency points. What is the length proportion of line segments x : y?

Categories
Intermediate

The wall between us

Two circles inscribed an a trapezium divided by a perpendicular line segment. What is the proportion of its length a and the side b?