# Thread: finding radius and center of circles inscribed into triangles

1. ## finding radius and center of circles inscribed into triangles

Hi everyone! I would like guidance on setting this problem up. I'm not actually in a geometry course so I am not sure where to begin.

Two triangles share a common edge. The vertices are located at the Cartesian coordinates of (1,10), (0.75, 15), (18,12), and (8,2). What is the radius and center location of the largest inscribed circle for each triangle? What is the distance between the two centers?

*If it helps, the common edge is between the coordinates (1,10) and (18,12).

I found a wiki article: Incircle and excircles of a triangle - Wikipedia, the free encyclopedia
But I don't know how to find the length of the sides. These are not right triangles.

If anyone can help with this in any way, I'd really appreciate it. Thanks in advance for your time.

2. ## Re: finding radius and center of circles inscribed into triangles

Originally Posted by headabovewater
Hi everyone! I would like guidance on setting this problem up. I'm not actually in a geometry course so I am not sure where to begin.

Two triangles share a common edge. The vertices are located at the Cartesian coordinates of (1,10), (0.75, 15), (18,12), and (8,2). What is the radius and center location of the largest inscribed circle for each triangle? What is the distance between the two centers?

*If it helps, the common edge is between the coordinates (1,10) and (18,12).

I found a wiki article: Incircle and excircles of a triangle - Wikipedia, the free encyclopedia
But I don't know how to find the length of the sides. These are not right triangles.

If anyone can help with this in any way, I'd really appreciate it. Thanks in advance for your time.

i really don't remeber how to solve it but i can help u with that drawing in attachment

3. ## Re: finding radius and center of circles inscribed into triangles

Originally Posted by mido22
i really don't remeber how to solve it but i can help u with that drawing in attachment
this is fantastic! very similar to my sketch, with circles inscribed into each triangle!

thanks for your contribution!

4. ## Re: finding radius and center of circles inscribed into triangles

Originally Posted by headabovewater
Hi everyone! I would like guidance on setting this problem up. I'm not actually in a geometry course so I am not sure where to begin.

Two triangles share a common edge. The vertices are located at the Cartesian coordinates of (1,10), (0.75, 15), (18,12), and (8,2). What is the radius and center location of the largest inscribed circle for each triangle? What is the distance between the two centers?

*If it helps, the common edge is between the coordinates (1,10) and (18,12).

I found a wiki article: Incircle and excircles of a triangle - Wikipedia, the free encyclopedia
But I don't know how to find the length of the sides. These are not right triangles.

If anyone can help with this in any way, I'd really appreciate it. Thanks in advance for your time.
1. The length of the radius of the incircle can be calculated by

$\rho = \dfrac{2A}{a+b+c}$

where $\rho$ = radius of the incircle (orange)
A = area of the triangle
a, b, c = lengthes of the sides of the triangle

2. The midpoint of an incircle is the point of intersection of two angle bisectors of the interior angles of the triangle (green). To avoid the (mostly) quite tricky calculations with angle bisectors you can use two parallels to sides of the triangle which have the distance $\rho$ from the sides.

3. I've attached a sketch so you can check the results of your calculations. The distance between the two midpoints is drawn in red.

5. ## Re: finding radius and center of circles inscribed into triangles

Originally Posted by earboth
1. The length of the radius of the incircle can be calculated by

$\rho = \dfrac{2A}{a+b+c}$

where $\rho$ = radius of the incircle (orange)
A = area of the triangle
a, b, c = lengthes of the sides of the triangle
this is helpful. thanks for your time.
i'm guessing the area would be calculated from the sides as well...but

how would i go about calculating the lengths of the sides? is it purely the distance of the coordinates? for example from 1, 10 to .8, 15, would that side just be 5?

i'm sorry if this is elementary as i mentioned, i am not taking a class that covers this.

6. ## Re: finding radius and center of circles inscribed into triangles

Originally Posted by headabovewater
this is helpful. thanks for your time.
i'm guessing the area would be calculated from the sides as well...but

how would i go about calculating the lengths of the sides? is it purely the distance of the coordinates? for example from 1, 10 to .8, 15, would that side just be 5?
If you have the points $P(x_P, y_P)$ and $Q(x_Q, y_Q)$ then the distance $|\overline{PQ}| = \sqrt{(x_P - x_Q)^2+(y_P-y_Q)^2}$.
One of the many applications of the Pythagorean theorem.

i'm sorry if this is elementary as i mentioned, i am not taking a class that covers this.
If the vertices of a polygon are determined by their coordinates then you can calculate the area by the chop-off method: You choose a rectangle whose sides are parallel to the axes and who contains completely the area in question (see attachment). Then you "cut off" those right triangles (or trapezods) which don't belong to the area. What's left is the value of the area.

With my example:
The surrounding rectangle has the area $a_r = 5 \cdot 6 = 30$
The areas of the triangles $\Delta_1 = 7.5, \Delta_2 = 4.5, \Delta_3 = 6$

Subtract the areas of the triangles from the area of the rectangle : 30 -18 = 12.

So the red-striped triangle has an area of 12 (units)

7. ## Re: finding radius and center of circles inscribed into triangles

Wow, I think I understand now! I will work on these calculations and return with my results.

Thanks so much!

Originally Posted by earboth
If you have the points $P(x_P, y_P)$ and $Q(x_Q, y_Q)$ then the distance $|\overline{PQ}| = \sqrt{(x_P - x_Q)^2+(y_P-y_Q)^2}$.
One of the many applications of the Pythagorean theorem.

If the vertices of a polygon are determined by their coordinates then you can calculate the area by the chop-off method: You choose a rectangle whose sides are parallel to the axes and who contains completely the area in question (see attachment). Then you "cut off" those right triangles (or trapezods) which don't belong to the area. What's left is the value of the area.

With my example:
The surrounding rectangle has the area $a_r = 5 \cdot 6 = 30$
The areas of the triangles $\Delta_1 = 7.5, \Delta_2 = 4.5, \Delta_3 = 6$

Subtract the areas of the triangles from the area of the rectangle : 30 -18 = 12.

So the red-striped triangle has an area of 12 (units)

8. ## Re: finding radius and center of circles inscribed into triangles

ok so i have finished this part of the problem. my radii match so far, and the sides were 5.006, 17.509, 11.662.

i now need to plot this and it's giving me HELL.....i have verified all statements with the help files, but i cant get the plots to print at all. earboth, what program did you use for the above plot?

should i post the plot part of the question to the other section of the board?

### afind coordinates of centre of circle inscribed in a triangle

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