with only have the points on a triangle how would you determine the radius and center of the incircle of that triangle?
There are 4 centres of a triangle,
the centroid, the orthocentre, the circumcentre and the incentre.
of those four you asked for the hardest one to find, lol
By definition, the incentre is the point in a triangle which is equidistant from each of the three sides
Are you familiar with the formula for finding the distance from a given point to a given line?
The distance from (p,q) to the line Ax + By + C = 0
= |Ap +Bq + C|/√(A^2+B^2)
so find the equation of each of the 3 sides of the triangle and write them in the general form
Ax+By+C=0
then using your given point, find 3 different expressions for the distance.
They must all be equal.
form 2 equations by setting any two of the expressions equal to each other.
That will give you 2 equations containing p and q.
solve.
Good luck, this is usually a very messy problem unless it was carefully planned.
(If somebody has a better way to do this, I would appreciate letting me know)
The general form of the equations are (4/3)x+y=(1688/3)
(-9/40)x+y=(19577/40)
(5/12)x+y=(-3929/12)
but then when I use the incenter(p,q)and express it for distance and set the equations equal to each other i get different things when I set the first 2 equations together than I do the second 2 and the first and last. Can you tell me what I am doing wrong?
To determine the radius and center of the incircle of a triangle, you need to know the coordinates of the three vertices. Let's assume the coordinates of the three vertices of the triangle are A(x1, y1), B(x2, y2), and C(x3, y3).
Here are the steps to find the radius and center of the incircle:
Step 1: Calculate the lengths of the sides of the triangle. You can use the distance formula to find the distance between two points. The lengths of the sides can be calculated as follows:
Side AB = √((x2 - x1)^2 + (y2 - y1)^2)
Side BC = √((x3 - x2)^2 + (y3 - y2)^2)
Side CA = √((x1 - x3)^2 + (y1 - y3)^2)
Step 2: Calculate the semi-perimeter of the triangle, which is defined as half the sum of the lengths of the sides:
Semi-perimeter = (Side AB + Side BC + Side CA) / 2
Step 3: Calculate the area of the triangle using Heron's formula:
Area of triangle = √(Semi-perimeter * (Semi-perimeter - Side AB) * (Semi-perimeter - Side BC) * (Semi-perimeter - Side CA))
Step 4: Calculate the inradius of the triangle using the formula:
Inradius = Area of triangle / Semi-perimeter
Step 5: Calculate the coordinates of the incenter. The incenter is the center of the incircle and can be calculated using the following formulas:
x-coordinate of incenter = ((Side BC * x1) + (Side CA * x2) + (Side AB * x3)) / (Side AB + Side BC + Side CA)
y-coordinate of incenter = ((Side BC * y1) + (Side CA * y2) + (Side AB * y3)) / (Side AB + Side BC + Side CA)
The coordinates obtained above will give you the center of the incircle.
Finally, you have the radius and center of the incircle of the triangle.