Synthetic Geometry
1. Triangle
- 1- simplex
1.1. Properties
1.1.1. Area Formula
- where
is the inradius, and is the semiperimeter.
- where
- where
is the circumradius.
- where
1.1.2. Inradius Formula
- from the Heron's formula
1.1.3. Circumradius Formula
2. Inverse Pythagorean Theorem
Reciprocal Pythagorean Theorem, Upside Down Pythagorean Theorem
2.1. Statement
3. Thales's Theorem
3.1. Statement
The inscribed angle subtended by the diameter of circle is right angle.
3.2. Inscribed Angle Theorem
- Inscribed angle is twice the central angle.
- Euclid's Elements, Proposition 20 on Book 3.
4. Ptolemy's Theorem
4.1. Statement
- For a cyclic quadrilateral
.
5. Apollonius' Theorem
- Special case of Stewart's theorem.
5.1. Statement
- For a triangle
and a middle point of the side :
6. Stewart's Theorem
- Relation between the lengths of the sides and the length of a cevian in a triangle.
6.1. Statement
- excalidraw:stweart.excalidraw
- If a cevian divides the side with length
into and :
6.1.1. Symmetric Form
- By introducing the signed length:
where are collinear points and is any point.
6.2. Mnemonic
- A man and his dad put a bomb in the sink.
7. Heron's Formula
7.1. Formula
7.2. Proof
- The area can be obtained in two ways:
- where
is the inradius, and is the semi-perimeter.
- where
- Using the triple cotangent identity:
- Using the triple cotangent identity:
- Multiplying two equations yields:
8. Parallelogram Law
8.1. Statement
- Given a parallelogram
:
8.1.1. In a Normed Space
9. Brahmagupta Theorem
9.1. Statement
10. Brahmagupta's Formula
- Special case of Bretschneider's formula.
10.1. Formula
- Area
of a cyclic quadrilateral with side lengths : - where
is the semiperimeter.
11. Bretschneider's Formula
- Area of a general quadrilateral both convex and concave, but not self-intersecting ones.
11.1. Formula
Figure 1: tetra
12. Napoleon's Theorem
12.1. Statement
The three center of the equilateral triangles subtended from three sides of any triangle forms an equilateral triangle.
13. Ceva's Theorem
13.1. Cevian
Line segment that joins a vertex and a point on the opposite side in a triangle.
13.2. Statement
- Given three cevians that intersects at one point as follows, not considering the degenerate cases:
13.3. Properties
- It is a theorem of affine geometry, that is true for any affine plane over any field.
14. Menelaus's Theorem
- The projective dual of the Ceva's theorem
14.1. Statement
- where the lengths are signed.
- The lengths can be interpreted as the homothety .
15. Pappus's Hexagon Theorem
15.1. Statement
- Given a pair of sets of collinear points,
, the intersection points are collinear on the Pappus line.
16. Pascal's Theorem
- hexagrammum mysticum theorem
- Generalization of Pappus's haxagon theorem.
16.1. Statement
- The three intersections of the three pairs of line segments that connects opposite points of any six points on a conic section is collinear.
- The line that the intersections passes is called the Pascal line of the hexagon.
16.2. Corollary
- Midpoints of parallel lines are collinear to the center.
- How to find the center, major axis and minor axis of an ellipse - Quora
17. Braikenridge-Maclaurin Theorem
- The converse of Pascal's theorem.
17.1. Statement
- If the three intersection points of the three pairs of lines through the opposite sides of a hexagon are collinear, then the vertices lie on a conic.
17.2. Corollary
- Any five points uniquely determines a conic.
18. Butterfly Theorem
18.1. Statement
- The two triangles subtended by the two chords passing through the midpoint (more generally, two points that are equidistant from the midpoint) of any chord, intersect with the original chord at the same distance from the midpoint.
- It is also applicable to any conic section.
19. Viviani's Theorem
19.1. Statement
The sum of the distance from any point in a equilateral triangle to each edge is equal to the height of the triangle.
19.2. Ternary Plot
A 2-dimensional plot that can represent three variables with constant sum.
- It make use of barycentric coordinate system, thus making it convenient to represent a composition.
19.2.1. Examples
- CIE xy chromaticity diagram
- Soil texture diagram
- Piper diagram
- Flammability diagram
20. Conway Circle Theorem
20.1. Statement
- After extending the sides at each vertex by the length of the opposite side, then the six points at the end of lines lie on a circle called Conway circle which is concentric to the incenter.
20.2. Properties
- The radius of the Conway circle is
, where is the inradius and is the semiperimeter. - This theorem is the special case of the windscreen wiper theorem, which states that by swiveling a point, one can obtain a circle.
21. Triangle Center
- ENCYCLOPEDIA OF TRIANGLE CENTERS
- Triangle center, central line, triangle conics, and triangle cubic is the central object in the modern study of triangle. They classifies the points based on their properties, for a given set of triangles.
21.1. Incenter
21.2. Centroid
21.3. Circumcenter
21.4. Orthocenter
21.5. Nine-Point Center
Center of the circle passing through
- the midpoint of each side
- the foot of each altitude
- and the midpoint between the orthocenter and each vertex.
The circle goes by various names: nine-point circle, Feuerbach's circle, Euler's circle, Terquem's circle
21.6. Symmedian Point
Intersection of the symmedians.
21.7. Gergonne Point
Intersection of the lines connecting each vertex to the point where the incircle touches the opposite side.
21.8. Nagel Point
Intersection of the lines connecting each vertex to the point where the excircle touches the opposite side.
21.9. Mittenpunkt
- Invariant point under affine transformation?
21.10. Spieker Center
Incenter of the medial triangle.
21.11. Feuerbach Point
The tangent point of the incircle and the nine-point circle
21.11.1. Feuerbach's Theorem
The nine-point circle is tangent to the incircle. As a remark, it is also tangent to the three excircles.
21.12. Fermat Point
22. Triangle Cubic
A triangle cubic
22.1. Neuberg Cubic
Locus of point
23. Cayley-Menger Determinant
- Generalization of Heron's formula
23.1. Definition
The content of an
Equivalently:
24. Reference
- Ptolemy's theorem - Wikipedia
- Apollonius's theorem - Wikipedia
- Stewart's theorem - Wikipedia
- A Miraculous Proof (Ptolemy's Theorem) - Numberphile - YouTube
- Heron's formula - Wikipedia
- Parallelogram law - Wikipedia
- Brahmagupta theorem - Wikipedia
- Brahmagupta's formula - Wikipedia
- Bretschneider's formula - Wikipedia
- Ceva's theorem - Wikipedia
- Menelaus's theorem - Wikipedia
- Pappus's hexagon theorem - Wikipedia
- Pascal's theorem - Wikipedia
- Braikenridge–Maclaurin theorem - Wikipedia
- Napoleon's theorem visual proof | mathocube | - YouTube
- Conway circle theorem - Wikipedia
- Butterfly theorem | The beauty of geometry - YouTube
- Conway's IRIS and the windscreen wiper theorem - YouTube
- Modern triangle geometry - Wikipedia
- Triangle center - Wikipedia
- Neuberg cubic - Wikipedia
- Cayley–Menger determinant - Wikipedia