A triangle containing an altitude and the inscribed circle. One tangency point is shown. The red line segments have equal length. What is the proportion of the triangle sides a : b : c?
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And counting
Tag: length
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Protection
Two coloured squares inside a triangle inside a square of side length a. What is the length proportion a : b?
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Wheels in motion
Two tangent circles and a third circle touching the line segment connecting their centres in the tangency point. Prove that the red points (centre and two intersection points) are collinear.
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It’s complicated
A circle, a semicircle and several triangles. The black points are tangency points. What proportions have the line segments a : r : b?
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Parallel universes
Two straight lines are specified by y=a+cx and y=b+cx, respectively, where b≥a. What’s the distance d between them in terms of a, b and c?
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The kingsize bed
A rectangle with an inscribed right triangle. What is ac/bd?
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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.
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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?