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Intercept theorem. The intercept theorem, also known as Thales's theorem, basic proportionality theorem or side splitter theorem, is an important theorem in elementary geometry about the ratios of various line segments that are created if two rays with a common starting point are intercepted by a pair of parallels.
Two lines (in black) meeting a triangle side internally and meeting the other sides internally and externally The axiom states that, [3] Pasch's axiom — Let A , B , C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A , B , C .
The triangle shown is the unique isosceles triangle for which there are exactly two perpendicular quadrisections. [1] In triangle geometry, the Bernoulli quadrisection problem asks how to divide a given triangle into four equal-area pieces by two perpendicular lines. Its solution by Jacob Bernoulli was published in 1687.
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.
The triangle DEF is called the pedal triangle of P. The antipedal triangle of P is the triangle formed by the lines through A, B, C perpendicular to PA, PB, PC respectively. Two points P and Q are called counter points if the pedal triangle of P is homothetic to the antipedal triangle of Q and the pedal triangle of Q is homothetic to the ...
Trilinear coordinates. In geometry, the trilinear coordinates x : y : z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio x : y is the ratio of the perpendicular distances from the point to the ...
In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle 's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle. Note that this theorem is not to be confused with the Inscribed ...
Perpendicular lines from the side midpoints (intersect at the circumcenter) In geometry, the Euler line, named after Leonhard Euler ( / ˈɔɪlər / ), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the ...