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The midpoint theorem generalizes to the intercept theorem, where rather than using midpoints, both sides are partitioned in the same ratio. [1] [2] The converse of the theorem is true as well. That is if a line is drawn through the midpoint of triangle side parallel to another triangle side then the line will bisect the third side of the triangle.
Given two points of interest, finding the midpoint of the line segment they determine can be accomplished by a compass and straightedge construction.The midpoint of a line segment, embedded in a plane, can be located by first constructing a lens using circular arcs of equal (and large enough) radii centered at the two endpoints, then connecting the cusps of the lens (the two points where the ...
By Girard's theorem, the sum of the angles of a triangle on a sphere is (+), where is the fraction of the sphere's area enclosed by the triangle. [ 73 ] [ 74 ] In more general spaces, there are comparison theorems relating the properties of a triangle in the space to properties of a corresponding triangle in a model space like hyperbolic or ...
Due to the Pythagorean theorem the number () has the simple geometric meanings shown in the diagram: For a point outside the circle () is the squared tangential distance | | of point to the circle . Points with equal power, isolines of Π ( P ) {\displaystyle \Pi (P)} , are circles concentric to circle c {\displaystyle c} .
This fact is known as the Feuerbach conic theorem. The nine point circle and the 16 tangent circles of the orthocentric system If an orthocentric system of four points A, B, C, H is given, then the four triangles formed by any combination of three distinct points of that system all share the same nine-point circle.
In Euclidean geometry, the intersecting chords theorem, or just the chord theorem, is a statement that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's Elements.