1) List the geometric properties of each triangle:
Equilateral Triangle
Scalene Triangle
Isosceles Triangle
Equilateral Triangle:
- All three sides are equal in length.
- All three angles are equal and measure 60 degrees.
- The medians, altitudes, and perpendicular bisectors coincide and are concurrent at the centroid.
- The circumcenter, orthocenter, and incenter are all the same point.
- The area can be calculated using the formula A = (sqrt(3)/4) * s^2, where s is the length of any side.
Scalene Triangle:
- All three sides have different lengths.
- All three angles have different measures.
- The medians divide each other into segments with a ratio of 2:1.
- The altitudes are concurrent at the orthocenter.
- The perpendicular bisectors are concurrent at the circumcenter.
- The incenter lies inside the triangle.
- The area can be calculated using Heron's formula: A = sqrt(s(s-a)(s-b)(s-c)), where s is the semiperimeter and a, b, c are the lengths of the sides.
Isosceles Triangle:
- Two sides are equal in length.
- The angles opposite the equal sides are also equal.
- The median, altitude, and perpendicular bisector drawn from the vertex opposite the equal sides all bisect the angle of the vertex.
- The circumcenter lies on the perpendicular bisector of the base.
- The incenter lies on the angle bisector of the vertex angle.
- The orthocenter lies on the altitude drawn from the vertex angle.
- The area can be calculated using the formula A = (b/4) * sqrt(4a^2 - b^2), where a is the length of the equal sides and b is the length of the base.