## The Dancing Triangle and Its Applications

In the figure below, lines l and m are parallel lines. What can you say about the areas of triangle ABC and triangle ADC?

The distance between two parallel lines is equal at any point, so the two triangles have the same altitude (can you see why?). Further, the two triangles have a common base, therefore, their base lengths are equal. So, the areas of the two triangles are equal. In fact, you can choose any point P on line l and the areas of the triangle ACP will always equal to the areas of triangles ABC and ADC. We like to call this triangle the dancing triangle because using an applet, you can dance it by moving P without changing the area. In the applet below, move points B and D to dance the triangle.  » Read more

## On Equations of Intersecting, Coinciding, and Parallel Lines

When we have two lines on a plane, there are three possibilities:

• the lines will never meet (parallel)
• the lines will meet at one point (intersecting)
• the lines will meet at infinitely many points (coinciding).

As for the third case, coinciding means that lines which are on top of each other.

In Algebra, we have learned that a line can be represented with an equation. The equations which represent lines are called linear equations. We have learned that linear equations can be represented by $y = mx + b$, where $m$ and $b$ are real numbers.

We can examine the three cases mentioned above in terms of equations. Can we determine if lines are parallel, intersecting, or coinciding based on equations only?  » Read more

## The 3 Saddest Mathematics Love Stories

There are many unfortunate events in mathematics as well as the lives of mathematicians. You haven’t probably heard the misfortunes of Evariste Galois; he died at the age of 20 in a duel for a lady who didn’t love him.You probably know some famous theorems by great mathematicians, but you probably didn’t know that they died young.  You probably do not know that Descartes could not marry either of the two woman he loves because they were on different social status.

But despite these unimaginable events, there are actually untold stories that history failed to acknowledge. I think they are probably the 3 saddest mathematics love stories ever told. Now, don’t be depressed.

The 3 Saddest Mathematics Love Stories

1. Tangent lines who had one chance to meet and then parted forever.

2. Parallel Lines who were never meant to meet.

3. Asymptotes who can get closer and closer to each other but will never be together.

Source: Unknown (I saw this in Facebook).