Some interesting things about polygons and triangles

Contents of this section: Polygons and Triangles Pythagorean Theorem Area Inscribed Angle Intersecting Chords - intercepted arcs Intersecting Chords - Product Theorem Diagonals of an n-gon Apollonius' Theorem Right Triangle Median, Altitude, Bisector Equilateral Triangle Crossed Ladders Isosceles Trapezoid Proof Steiner-Lehmus Theorem Cyclic Quadrilateral Rectangles, Squares Quadrilaterals FTDSF Urquhart's Theorem Triangle Centers Barycentric Coordinates Ceva's Theorem Carnot's Theorem Cross Ratio Theorem Isogonal Conjugates Triangle Trisection Trisectrix

Pythagorean Theorem gives a visual proof of the famous theorem: the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the legs.

Vector Area gives a way of calculating the area of a triangle formed by two vectors.

Heron's Formula gives the area of a triangle given the lengths of its sides.

The various centers of a triangle, such as the incenter, circumcenter, and orthocenter are explored in this section.  Check out these proofs: The radius of an inscribed circle is r = K/s, and the radius of a circumscribed circle is R = abc/(4K),
where the lengths of the sides are a, b, and c, the semiperimeter s=(a+b+c)/2, and the area is K.

Internet references

The Nine-point circle, from mathworld, is the circle that passes through the "feet" of the three altitudes of a triangle.  It also passes through the midpoints of the three sides, and the three midpoints of the segments connecting the vertices to the orthocenter; hence "nine points".

Related pages in this website

Point and Triangle gives a clever method of telling whether a point is inside a triangle.  An algorithm to do this is written in several different languages.

The Triangle Inequality proves that |a+b|<=|a|+|b| and |a-b|>=|a|-|b|

Triangle Circles -- Inscribed Circle and Angle Bisectors, with bonus features: exscribed circles and nine-point circle

The webmaster and author of this Math Help site is Graeme McRae.