25
votes
Accepted
What is the most triangles you can make from a capital "H" and 3 straight lines?
Here's a solution for 7 triangles:
- 78.6k
14
votes
Accepted
Prove why this mechanical linkage for a triangle centroid works
The proof is in two parts, corresponding to the two linkages which are joined to each other at a single point. For each part, I'll try to both explain in words and illustrate on the picture you've ...
- 117k
12
votes
Accepted
What percentage of blue?
Total number of triangles:
w : white half parallelograms
b : blue half parallelograms
So...
Image:
- 2,188
11
votes
Accepted
10
votes
Accepted
Professor Rackbrane: Count the triangles
We distinguish the triangles by how many of the short sides (ABCDEA) they use:
- 7,820
9
votes
8
votes
3D? No-no! 3 Sides
Here is the solution to the puzzle (note the correction in "3,1,2,5" to "3,1,2,1,4", by comment here):
- 5,191
8
votes
Accepted
7
votes
Accepted
7
votes
Accepted
Find the least expense?
Assuming "transportation cost" means sum of distances to each of the three roads, and the side of the equilateral triangle has length $1$:
- 18.1k
7
votes
Accepted
Special triangles in convex polygons
Here is a convex dodecagon made of $50$ of those triangles.
Can it be done with fewer?
- 55k
7
votes
Accepted
Circle inscribed in triangle problem
Let's draw a few more points and line segments:
By looking at the side lengths,
Therefore, angle $BEC$ is
By quadrilateral $APEQ$, the angle $PEQ$ is $180-22=158$ degrees,
Commentary
Originally (...
- 117k
6
votes
Accepted
3D? No-no! 3 Sides
I was trying to post this 5 minutes before the other answer, but got snookered by camp wifi
- 3,751
6
votes
What is the most triangles you can make from a capital "H" and 3 straight lines?
Here's one with six triangles (7 if you count triangles outside of triangles, which you don't):
- 9,337
6
votes
Professor Rackbrane: Count the triangles
I have a general method for counting triangles in an given figure.
- 31k
5
votes
Accepted
A triangle inside a triangle
The answer is
because, for example, if T2 has side-lengths
then a triangle T1 such as
More generally, we can consider T2 with side-lengths
and T1 with side-lengths
- 117k
5
votes
Accepted
5
votes
Accepted
Triangles to diamonds
As Bubbler already noted in a comment, the final formula is
To derive that, I will use the following facts and properties.
Triangle areas
In-radius
Rhombus
Now let's put all this together:
- 55k
5
votes
Special triangles in convex polygons
It is possible to do better than a hexagon, if an irregular polygon is acceptable.
It is also possible to construct an equilateral triangle or a hexagon. On reflection (and thanks to @Hugh's comment) ...
- 14k
5
votes
Accepted
4
votes
Accepted
Two difficult "Seventeen right isosceles triangles into a square" tilings
Here are the solutions to both questions:
- 13.6k
4
votes
Accepted
Five graded difficulty isosceles right triangle into square tilings
Here are the solutions to the five problems.
- 13.6k
4
votes
Accepted
3
votes
Prove why this mechanical linkage for a triangle centroid works
Explanation
Proof sketch of Centroid Theorem (this is a good geometry puzzle)
- 989
3
votes
What is the most triangles you can make from a capital "H" and 3 straight lines?
Does this count as 8 triangles?
- 767
3
votes
Accepted
20 right isosceles triangles into a square
Here are at least two solutions (up to reflection and rotation)
- 13.6k
3
votes
3
votes
Accepted
How to map barycentric indices to a single integer?
[Note: OP has revised the question with a slightly different barycentric indexing scheme, which makes the following no longer quite right. I'll fix it up in a minute.]
Number the rows of triangles ...
- 121k
3
votes
3
votes
Professor Rackbrane: Count the triangles
Here is a slightly different solution which focuses on vertices:
- 15.6k
Only top scored, non community-wiki answers of a minimum length are eligible