Inscribed and Circumscribed Circles of a Triangle

This page introduces the concepts of okrąg wpisany w trójkącie (inscribed circle) and okrąg opisany na trójkącie (circumscribed circle) for different types of triangles. It provides formulas for calculating the radii of these circles and related triangle properties.

Definition: The inscribed circle of a triangle is the largest circle that can be drawn inside the triangle, touching all three sides. The circumscribed circle passes through all three vertices of the triangle.

Highlight: The center of the inscribed circle is located at the intersection of the angle bisectors of the triangle.

For a trójkąt prostokątny (right triangle), the following formulas are provided:

• Area: P = ah/2, where 'a' is the base and 'h' is the height
• Radius of the circumscribed circle: R = c/2, where 'c' is the hypotenuse

Example: In a right triangle with legs 3 and 4, and hypotenuse 5, the radius of the circumscribed circle would be R = 5/2 = 2.5.

For a general triangle, the radius of the inscribed circle is given by:

r = P / s, where P is the area and s is the semi-perimeter

The semi-perimeter is calculated as: s = (a + b + c) / 2, where a, b, and c are the side lengths of the triangle.

For a trójkąt równoramienny (isosceles triangle), specific formulas are provided for the radius of the circumscribed circle and the height:

• R = a² / (4h), where 'a' is the length of the equal sides and 'h' is the height
• h = a√3 / 2, relating the height to the side length

Vocabulary: The height (h) in an isosceles triangle is the perpendicular line segment from the vertex angle to the base.