At around timestamp 4:30, why is the height line where it is? Why doesn't it start at the intersection in the middle (the 120 degree angle?)
How does Prof. Loh know that all the triangles have the same area?
Hi TSS Graviser,
This is a really interesting and surprising characteristic of parallelograms, that the four triangles formed from the intersection of their diagonals have the same area! I'm glad you asked about it.
For your first question, actually, there are three ways of thinking of a base and height for a given triangle. This is because there are three sides of a triangle! In the diagram below, I have drawn the three different heights for the yellow triangle, as well as the three different heights for the purple triangle. I tried to color-code the heights with their matching bases.
The idea is that a height must be perpendicular to the base. That means an altitude might have to lie outside the triangle, which is true for obtuse triangles.
Now, for the four triangles formed from the diagonals of the parallelogram, let's look at this picture:
The purple triangle and yellow triangles have the same altitude, which is labeled as "h." We are choosing here the red line to be the base of the purple triangle, and the green line to be the base of the yellow triangle. Note that the line marked with an \(h\) is indeed perpendicular to both of these base. Now, the exciting part is that, due to the symmetry of the parallelogram, the diagonals cut each other in half, so the red line is the same length as the green line. This is why the area of the purple triangle is equal to the area of the yellow triangle.
The other two triangles which were not colored in the diagram are just flipped versions of these two, so the same applies to them, and any such triangles drawn from any parallelogram's diagonals.
The Daily Challenge Team