HW #3 Circumcenter & Incenter

Front

What is the circumcenter of a triangle?

Back

The circumcenter of a triangle is the point where the perpendicular bisectors of the sides intersect. It is equidistant from all three vertices of the triangle.

Front

How do you find the circumcenter of a triangle given its vertices?

Back

To find the circumcenter, calculate the midpoints of at least two sides of the triangle, then find the slopes of those sides. Use the negative reciprocal of the slopes to find the equations of the perpendicular bisectors, and solve the equations simultaneously to find the circumcenter.

Front

What is the incenter of a triangle?

Back

The incenter of a triangle is the point where the angle bisectors of the triangle intersect. It is equidistant from all three sides of the triangle.

Front

How do you find the incenter of a triangle given its vertices?

Back

To find the incenter, calculate the lengths of the sides of the triangle, then use the formula: I = (aA + bB + cC) / (a + b + c), where A, B, and C are the vertices and a, b, and c are the lengths of the sides opposite those vertices.

Front

What is the relationship between the circumcenter and the circumradius?

Back

The circumradius is the radius of the circumcircle, which is the circle that passes through all three vertices of the triangle. The circumcenter is the center of this circumcircle.

Front

What is the formula for the circumradius (R) of a triangle?

Back

The circumradius R can be calculated using the formula: R = (abc) / (4K), where a, b, and c are the lengths of the sides of the triangle and K is the area of the triangle.

Front

What is the significance of the circumcenter in triangle geometry?

Back

The circumcenter is significant because it helps in constructing the circumcircle, determining the circumradius, and is used in various geometric proofs and constructions.

Front

What is the formula for the area (K) of a triangle using its sides?

Back

The area K can be calculated using Heron's formula: K = √[s(s-a)(s-b)(s-c)], where s is the semi-perimeter (s = (a+b+c)/2) and a, b, c are the lengths of the sides.

Front

What is the relationship between the incenter and the inradius?

Back

The inradius is the radius of the incircle, which is the circle that is tangent to all three sides of the triangle. The incenter is the center of this incircle.

Front

What is the formula for the inradius (r) of a triangle?

Back

The inradius r can be calculated using the formula: r = K / s, where K is the area of the triangle and s is the semi-perimeter.