Tag Archives: plane geometry

Area vs. Perimeter Puzzle

This surprising, but simple, puzzle is from the 12 April MathsMonday offering by MEI, an independent curriculum development body for mathematics education in the UK.

“In the diagram various regular polygons, P, have been drawn whose sides are tangents to a circle, C.  Show that for any regular polygon drawn in this way:”

(Given that the polygons approximate the circle in the limit, it would not be surprising that this relationship would hold—in the limit.  It is surprising that it should be true for every regular polygon that circumscribes the circle.)

See the Area vs. Perimeter Puzzle

Geometric Puzzle Magnificence

Here is yet another collection of beautiful geometric problems from Catriona Agg (née Shearer).  For some reason I found these a bit more challenging than the previous ones.   Some of them required more time to “see” the breakthrough.

See Geometric Puzzle Magnificence (revised)

(Update 11/13/2022)  Problem #3 Solution Corrected Continue reading →

Equitable Slice Problem

This is another Brainteaser from the Quantum math magazine .

“How can a polygonal line BDEFG be drawn in a triangle ABC so that the five triangles obtained have the same area?”

I found this problem rather challenging, especially when I first tried to solve it analytically (using hyperbolas).  Eventually I arrived at a procedure that would accomplish the result. (revised)

See the Equitable Slice Problem  (revised)

(Update 9/22/2021)  I goofed.  I erroneously and foolishly thought Quantum had not solved the problem.  Upon a closer reading I see what they were getting at and revised the posting.

Three Equal Circles

Here is a problem from the Quantum magazine, only this time from the “Challenges” section (these are expected to be a bit more difficult than the Brainteasers).

“Three circles with the same radius r all pass through a point H.  Prove that the circle passing through the points where the pairs of circles intersect (that is, points A, B, and C) also has the same radius r.”

Indeed, I found this quite challenging.  It took me several weeks to work out my approach and details.

See Three Equal Circles

Twin Intersection Puzzle

This is an interesting problem from the 1977 Canadian Math Society’s magazine, Crux Mathematicorum.

“206. [1977: 10] Proposed by Dan Pedoe, University of Minnesota.

A circle intersects the sides BC, CA and AB of a triangle ABC in the pairs of points X, X’, Y, Y’ and Z, Z’ respectively. If the perpendiculars at X, Y and Z to the respective sides BC, CA and AB are concurrent at a point P, prove that the respective perpendiculars at X’, Y’ and Z’ to the sides BC, CA and AB are concurrent at a point P’.”

See the Twin Intersection Puzzle

Puzzles and Problems: plane geometry, Dan Pedoe, Crux Mathematicorum

Shy Angle Problem

Here is yet another problem from Presh Talwalkar. This one is rather elegant in its simplicity of statement and answer.

“Solve For The Angle – Viral Puzzle

I thank Barry and also Akshay Dhivare from India for suggesting this problem!  This puzzle is popular on social media. What is the measure of the angle denoted by a “?” in the following diagram? You have to solve it using elementary geometry (no trigonometry or other methods).  It’s harder than it looks.  I admit I did not solve it. Can you figure it out?”

Answer.

See the Shy Angle Problem for solutions.

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