Essential Geometric Concepts: Triangles and Circles
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Classification of Triangles
| Classification by Sides | Classification by Angles |
|---|---|
| Equilateral: All 3 sides are equal. | Acute (Acutangle): Has 3 acute angles (less than 90°). |
| Isosceles: Has at least 2 equal sides. | Obtuse (Obtusangle): Has 1 obtuse angle (greater than 90°). |
| Scalene: All 3 sides are unequal (different lengths). | Right (Rectangle): Has 1 right angle (exactly 90°). |
Notable Lines and Points in a Triangle
- Altitude (Height): A segment drawn from a vertex perpendicular to the opposite side.
- Orthocenter: The point of intersection of the 3 altitudes of a triangle.
- Median: A line segment connecting a vertex to the midpoint of the opposite side.
- Centroid: The point of intersection of the 3 medians of a triangle.
- Angle Bisector: A line segment originating from a vertex that divides the corresponding interior angle into two equal parts.
- Incenter: The point of intersection of the 3 angle bisectors of a triangle.
- Perpendicular Bisector: A line perpendicular to a side passing through its midpoint.
- Circumcenter: The point of intersection of the 3 perpendicular bisectors of a triangle.
Elements and Properties of the Right Triangle
Elements of the Right Triangle
- Legs (Cathetus): The two sides adjacent to the right angle.
- Hypotenuse: The side opposite the right angle. It is always the longest side of the right triangle.
Key Properties of the Right Triangle
- The sum of the two acute angles is 90° (the acute angles are complementary).
- The legs serve as two of the triangle's altitudes. Consequently, the orthocenter coincides with the vertex of the right angle.
- Pythagorean Theorem: States that the square of the hypotenuse equals the sum of the squares of the lengths of the two legs ($a^2 + b^2 = c^2$).
- The circumcenter is located exactly at the midpoint of the hypotenuse.
The orthocenter, the centroid, and the circumcenter of any triangle are always collinear (lie on the same straight line). This line is known as the Euler Line. In an isosceles triangle, the incenter is also located on this line.
Fundamental Concepts of Circles
A Circle is the locus of all points in a plane that are equidistant from a fixed central point (the center).
A Disk (or Area of a Circle) is the portion of the plane bounded by the circle.
Key Elements of a Circle
- Radius: The distance from the center to any point on the circle.
- Arc of the Circle: A continuous portion of the circumference.
- Chord: A line segment determined by two points on the circle.
- Diameter: The chord that passes through the center of the circle. Its length is twice the radius ($D = 2r$).
- Semicircle (Arc): The arc determined by the diameter (half of the circumference).
- Semi-disk (Area): The portion of the plane bounded by a semicircle and the diameter that subtends it.
Parts of the Disk (Circular Regions)
- Circular Sector: The portion of the disk bounded by two radii and the arc between them.
- Circular Segment: The portion of the disk bounded by a chord and the arc it subtends.