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Geometry Theorems
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Pythagorean Theorem
In a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). i.e., .
Triangle Inequality Theorem
For any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Area of a Triangle
The area (A) of a triangle with base (b) and height (h) is given by .
Circumference of a Circle
The circumference (C) of a circle with radius (r) is given by .
Area of a Circle
The area (A) of a circle with radius (r) is given by .
Euler's Polyhedron Formula
For any convex polyhedron, the number of vertices (V), faces (F), and edges (E) satisfy the relationship .
Central Angle Theorem
In a circle, the central angle subtended by two points on the circle is twice the inscribed angle subtended by those points.
Parallel Postulate
Given a line and a point not on the line, there is exactly one line parallel to the given line that passes through the point.
Ceva's Theorem
In a triangle, three cevians (lines drawn from each vertex to the opposite side) are concurrent (intersect at a single point) if and only if , where the points D, E, and F lie on the sides of the triangle.
Angle Bisector Theorem
The angle bisector of an angle in a triangle divides the opposite side into two segments that are proportional to the lengths of the other two sides of the triangle.
Stewart's Theorem
For a triangle with sides a, b, and c and a cevian of length d dividing side a into segments of length m and n, Stewart’s Theorem is given by .
Heron's Formula
The area (A) of a triangle with sides a, b, and c and semi-perimeter (s) is given by , where .
Law of Cosines
In any triangle with sides of length a, b, and c, and the angle opposite of side c being gamma (), the Law of Cosines states .
Law of Sines
For any triangle with sides a, b, c and angles alpha (), beta (), and gamma (), the Law of Sines states .
Menelaus's Theorem
Given a triangle ABC and a transversal line intersecting the extended sides AB and AC at points L and M, and intersecting side BC at N, then Menelaus's Theorem states that the points L, M, and N are collinear if and only if .
Carnot's Theorem
For any triangle ABC, with the respective lengths opposite to the angles being a, b, and c, and the length from each vertex to the circumcenter being , , and , Carnot's Theorem is the condition .
Ptolemy's Theorem
For a quadrilateral inscribed in a circle (cyclic quadrilateral), if the lengths of the sides are a, b, c, and d, and the lengths of the diagonals are e and f, then Ptolemy’s Theorem states .
Desargues' Theorem
Two triangles are in perspective axially if and only if they are in perspective centrally; which means that the corresponding sides intersect in collinear points, or their extensions do.
Fermat's Point Theorem
For any triangle that is not equilateral, there is a point such that the total distance from the three vertices of the triangle to this point is minimized. This point is known as Fermat's point or Torricelli point.
Napoleon's Theorem
If equilateral triangles are constructed on the sides of any triangle (either all outwardly or all inwardly), the centers of those equilateral triangles themselves form an equilateral triangle.
Morley's Theorem
In any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, known as the Morley triangle.
Butterfly Theorem
Given a circle with a chord PQ and a midpoint M, if two other chords AB and CD pass through M and intersect PQ at X and Y respectively, then M is the midpoint of XY.
Nine Point Circle Theorem
There exists a circle in any non-degenerate triangle that passes through the midpoint of each side, the foot of each altitude, and the midpoint of the line segment from each vertex to the orthocenter.
Simson Line Theorem
Given a triangle ABC, and a point P on its circumcircle, the feet of the perpendiculars dropped from P to the sides of ABC (or their extensions) are collinear, and this line is known as the Simson line of point P.
Orthocenter
The orthocenter of a triangle is the point where the three altitudes, or the lines extended from them, intersect.
Incenter
The incenter of a triangle is the point where the three angle bisectors of the triangle intersect, and it is the center of the inscribed circle (incircle).
Circumcenter
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect, and it is the center of the circumscribed circle (circumcircle).
Centroid
The centroid of a triangle is the point where the three medians of the triangle intersect, and it serves as the triangle's center of mass.
Euler Line
In every non-equilateral triangle, the orthocenter (H), centroid (G), and circumcenter (O) are collinear, and the centroid is between the orthocenter and the circumcenter on this line, known as the Euler line.
Miquel's Theorem
In a triangle with three points, each lying on one side of the triangle, the circumcircles of the triangles formed with these points and the original triangle's vertices concur at a single point, known as Miquel's point.
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