CTK Insights

The three angle bisectors of a triangle meet at incenter of the triangle. Reversing the problem we may ask a relevant question: Given three concurrent lines: α, β, and γ. Is there always a triangle with the three lines as the angle bisectors. If so, construct the triangle. Solution Given three concurrent lines: α, β, [...]

The medians of a triangle meet at a point known at the center of the triangle. Reversing the problem we may ask a relevant question: Given three concurrent lines: α, β, and γ. Is there always a triangle with the three lines as the medians. If so, construct the triangle. Solution Given three concurrent lines: [...]

The altitudes of a triangle concur at a point - the orthocenter of the triangle. There are multitudes of proofs, each shedding light of a different hue on the existence of the orthocenter. Collecting these proofs was an enjoyable undertaking, and edifying, too. Not until a few days ago, when I came across another problem, [...]

Imagine a pyramid with no symmetries or regularities whatsoever. To construct a pyramid like that, pick a plane, four arbitrary points in the plane and one point outside. The lines (or rays) joining the latter to the four points in the plane serve as the edges of a slanted and likely irregular pyramid. However, the [...]

A problem for the innocent minds There is a well known problem of finding two points of the same color in the plane all points of which have been colored either red or blue distance 1 unit apart. It is simple but not trivial. This said what about finding two points of different colors under [...]

There are mathematical statements that appear counterintuitive. For example, when it comes to infinities, counterintuitive statements are abundant. At the other extreme, there are statements intuitively obvious that are rather hard to prove. Such, for example, is the famous Jordan Curve Theorem. Naturally, mathematics does not lack in statements of any intermediate kind. Here is [...]

This is a beautiful pieces by Andy Liu, University of Alberta, from the College Mathematics Journal, Volume 42, Number 5, November 2011, p. 372 Parallel lines are usually defined as lines with no points in common. Parallelism is clearly symmetric. If line 1 has no points in common with line 2, then line 2 also [...]

This puzzle comes from a wonderful Russian site, where its solution is presented as a sequence of animations. Is it possible to wrap the cube with a 3×3 piece of paper below it? Handling of the paper is subject to two conditions: The paper may be only cut or folded along the crease lines. The [...]

Parabola has an easily verifiable property. The segment joining points and crosses -axis in point . The equation of the segment is , from which . This may be a curious fact in its own right. What does it say? Taken at a face value, it simply shows a way to obtain the product of [...]

In a recent post, I have implied that Socrates new how to dissect a 2×1 rectangle into a square. There is actually no evidence that he did. However, he certainly knew how to produce a square half the area of a given one. How would he relate the two problems? A sangaku tablet has preserved [...]