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Synthetic Geometry

1. Triangle

1.1. Properties

1.1.1. Area Formula

  • A=rs
    • where r is the inradius, and s is the semiperimeter.
  • A=abc4R
    • where R is the circumradius.

1.1.2. Inradius Formula

1.1.3. Circumradius Formula

  • R=abc4s(sa)(sb)(sc)

2. Inverse Pythagorean Theorem

Reciprocal Pythagorean Theorem, Upside Down Pythagorean Theorem

2.1. Statement

Inverse_pythagorean_theorem.svg

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 ABCD. ABCD+BCAD=ACBD.

5. Apollonius' Theorem

5.1. Statement

  • For a triangle ABC and a middle point D of the side BC: 2(AB2+AC2)=BC2+4AD2.
12AB2+12AC2=14BC2+AD2BC2AB2+BC2AC2=BCBC24+BCAD2.

6. Stewart's Theorem

  • Relation between the lengths of the sides and the length of a cevian in a triangle.

6.1. Statement

6.1.1. Symmetric Form

  • By introducing the signed length: (PA2BC)+(PB2CA)+(PC2AB)+(ABBCCA)=0 where A,B,C are collinear points and P is any point.

6.2. Mnemonic

  • man+dad=bmb+cnc
  • A man and his dad put a bomb in the sink.

7. Heron's Formula

7.1. Formula

A=s(sa)(sb)(sc), where s is the semi-perimeter of a triangle.

7.2. Proof

  • The area can be obtained in two ways:
    • A=rs
      • where r is the inradius, and s is the semi-perimeter.
    • A=r(sa)+r(sb)+r(sc),
  • Multiplying two equations yields: A2=s(sa)(sb)(sc).

8. Parallelogram Law

8.1. Statement

  • Given a parallelogram ABCD: 2AB2+2BC2=AC2+BD2.

8.1.1. In a Normed Space

  • 2x2+2y2=x+y2+xy2.

9. Brahmagupta Theorem

9.1. Statement

Brahmaguptra's_theorem.svg

  • BMAC,EFBC|AF|=|FD|

10. Brahmagupta's Formula

10.1. Formula

  • Area K of a cyclic quadrilateral with side lengths a,b,c,d: K=(sa)(sb)(sc)(sd)
  • where s is the semiperimeter.

11. Bretschneider's Formula

  • Area of a general quadrilateral both convex and concave, but not self-intersecting ones.

11.1. Formula

Tetragon_measures.svg

Figure 1: tetra

  • K=(sa)(sb)(sc)(sd)abcdcos2(α+γ2)

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:

Ceva's_theorem.svg

AFFBBDDCCEEA=1, using the signed length.

13.3. Properties

  • It is a theorem of affine geometry, that is true for any affine plane over any field.

14. Menelaus's Theorem

14.1. Statement

Menelaus'_theorem.svg

  • AFFBBDDCCEEA=1.
  • where the lengths are signed.
  • The lengths can be interpreted as the homothety .

15. Pappus's Hexagon Theorem

15.1. Statement

Pappus-proj-ev.svg

  • Given a pair of sets of collinear points, ({A,B,C},{a,b,c}), the intersection points X,Y,Z are collinear on the Pappus line.

16. Pascal's 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

17. Braikenridge-Maclaurin 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.

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 r2+s2, where r is the inradius and s 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

X1, I

21.2. Centroid

X2, G

21.3. Circumcenter

X3, O

21.4. Orthocenter

X4, H

21.5. Nine-Point Center

X5, N

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

X6, K

Intersection of the symmedians.

21.7. Gergonne Point

X7, Ge

Intersection of the lines connecting each vertex to the point where the incircle touches the opposite side.

21.8. Nagel Point

X8, Na

Intersection of the lines connecting each vertex to the point where the excircle touches the opposite side.

21.9. Mittenpunkt

X9, M

  • Invariant point under affine transformation?

21.10. Spieker Center

X10, Sp

Incenter of the medial triangle.

21.11. Feuerbach Point

X11, F

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

X13, X

22. Triangle Cubic

A triangle cubic f(x,y,z)=0 is a cubic equation in the barycentric coordinates for a triangle.

22.1. Neuberg Cubic

Locus of point P in the plane of the reference triangle ABC such that, if the reflections of P in the sidelines of triangle are Pa,Pb,Pc, then the lines APa,BPb,CPc are concurrent.

23. Cayley-Menger Determinant

23.1. Definition

The content of an n-simplex formed by n+1 points is given as: vn2=1(n!)22n|2d012d012+d022d122d012+d0n2d1n2d022+d012d2122d022d022+d0n2d2n2d0n2+d012dn12d0n2+d022dn222d0n2| where dij is the distance between i-th and j-th points.

Equivalently: vn2=(1)n+1(n!)22n|0d012d022d0n21d1020d122d1n21d202d2120d2n21dn02dn12dn220111110|.

24. Reference

Created: 2025-05-13 Tue 01:31