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.
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The freight elevator
Tag: length
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The augmented chord
A circle with four chords. What is the length of line segment x in terms of lengths a and b?
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Fun with flags
A triangle with two cevians. What is d/c-b/a?
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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?
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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.
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The tornado
A quadrilateral covers exactly half of the area of a rectangle. Find the length x in terms of a, b and c.
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The steam iron
The midpoints of the hypotenuse of a right triangle and a line segment are connected. What is the length x?
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The major chord
A circle with four chords. What is the segment length x in terms of lengths a, b and c?
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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?
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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?