Tag Archives: plane geometry

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.

Down With Geometry

One of my favorite bloggers, Kevin Drum, decided to relieve the tedium of our current political anarchy by whacking the hornets’ nest of the high school mathematics curriculum, in particular the subject of plane geometry.  You can tell from the tag list on my blog that I hold plane geometry in high regard and can’t let this gibe pass without some rebuttal, futile as it may be.  Actually, I am not going to weigh in on the general issue of the current math curriculum that much, but rather make a few observations from my own experience over the years as it relates to Kevin’s post. 

See Down With Geometry

(Update 2/9/2021)   Vindication!  Continue reading →

Curve Making Puzzle

Here is a problem from Five Hundred Mathematical Challenges that I indeed found quite challenging.

“Problem 235. Two fixed points A and B and a moving point M are taken on the circumference of a circle. On the extension of the line segment AM a point N is taken, outside the circle, so that lengths MN = MB. Find the locus of N.”

Since one of the first hurdles I faced with this problem was trying to figure out what type of shape was being generated, I thought I would omit my usual drawings illustrating the problem statement.  There turned out to be a lot of cases to consider, but the result was most satisfying.  I also included the case when N is inside the circle.  Again Visio was my main tool to handle all the examples with the concomitant requirement to prove whatever Visio suggested.

See the Curve Making Puzzle

Circle-Halving Zigzag Problem

This is another delightful Brainteaser from the Quantum math magazine.

“All the vertices of a polygonal line ABCDE lie on a circumference (see the figure), and the angles at the vertices B, C, and D are each 45°.

Prove that the area of the blue part of the circle is equal to the area of the yellow part. (V. Proizvolov)”

I especially liked this problem since I was able to find a solution different from the one given by Quantum.  Who knows how many other variations there might be.

See the Circle-Halving Zigzag Problem

  1. Pages:
  3. 1
  4. 2
  5. 3
  6. 4
  7. 5
  8. 6
  9. 7
  10. 8
  11. 9
  12. 10
  13. 11
  14. 12
  15. 13