Real Life Applications - Circumcenter and Incenter

In this post, we will be looking at a couple of real life scenarios where having the knowledge about the Circumcenter and the Incenter might help us with a solution.

Applications of Circumcenter

There are two ways to look at the Circumcenter :

• As the point of intersection of three perpendicular bisectors of a triangle. 
• As the point which is equidistant from the three corners(vertices) of a triangle. 

Read more about the Circumcenter here.

Generally speaking, when you want to know the spot or location which is just about at an equal distance from three (non-linear) objects, you are basically looking for the circumcenter.

Let’s say there are three villages in the vicinity of each other and you want to open a fertiliser plant near them. The plant can be set up at a place such that it is approximately same distance away from the villages but close to all three villages as much as possible. 

Here the location of the circumcenter can be a perfect spot. If three dots(non-collinear) are representing the three main entrances to three villages, the circumcenter will be the point equidistant from all the three dots.

In a cold wintry December, a school in a city has organised a picnic trip to a water park, which is approximately two hours away from town if travelling by bus. 

But the problem is that there are too many students who have signed up for the trip, and the fact that the picnic spot is a long distance away, it would take a lot of time for a bus or buses to pick up each and every student from their respective homes, which will also result in higher fuel consumption. To save both fuel and time, the Circumcenter can help us here. 

We can divide students into groups such that each group comprises of three students who stay in about close proximity of each other. For each group of students the bus will halt at only one location and that location can be about the same distance from their three respective homes. So in this way, the total number of halts will be reduced and we will save time.

Here’s one more.....

There are some highly popular rides in an amusement park and then there are also less popular ones. Suppose three rides in the latter category are located near each other. As an economical manager of the park, instead of having a separate ticket selling kiosk near each of those less popular rides, you think that a single ticket selling kiosk for all three of them will be sufficient and will help reduce the spending. 

Again, the location for the single ticket selling kiosk can be chosen such that it is near about the same distance away from the three rides, i.e. at near about the circumcenter of the triangle with the three rides imagined to be its vertices. 

Applications of Incenter

Just like the Circumcenter, the Incenter of a triangle can be looked at in two ways :

• As the point of intersection of three angle bisectors of a triangle.
• As the point which is at the same distance from the three edges of a triangle.

Note that when we say the distance of the Incenter from an edge of a triangle we are talking about the line segment from the incenter that is perpendicular to the edge. In other words, drop three perpendicular line segments from the incenter on the three edges of a triangle, they will all be of the same length.

Read more about the Incenter here.

Imagine that we have a big dining table with its top surface in the shape of a triangle. Say we want to keep a water jug or a fruit tray at the “center” of the table such that it is easily and equally accessible to people from all three sides.

In order for the jug or any other item to be equally reachable from all three sides, its position on the table should be from where it is about the same distance away from the three sides. We can place it at or near about the location of the Incenter of the triangle.

Let’s look at another interesting possible situation where finding the incenter may solve our problem.

Say under a Government’s scheme, new concrete roads are been constructed in your village. There are total of three roads together forming the shape of a triangle surrounding your village.

Sensing a business opportunity, let’s say you want to open a small hotel somewhere inside this triangle formed by the roads. You would want to setup your business at a spot from where it can attract customers from all sides(roads) and must be equally reachable to people from all sides. Here the ground at or near the location of the incenter of the triangle could be an ideal location.

Cool fact : Incircle is also in the symbol for the deathly hallows. That’s ‘em legendary magical objects in J.K. Rowling’s wizarding world, aka Harry Potter universe. 

The triangle represents the undamageable, achromatic invisibility cloak, the vertical line inside the triangle depicts the unbeatable wand(magical stick). The inscribed (in-)circle represents a ring with the resurrection stone fitted into it, which (sort of) resurrects the dead. Whoever is in possession of the three hallows becomes the master of the death.

Read : Medians and the Centroid of a Triangle