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 Pascal's Mystic Hexagram Theorem Proof. Level: High School, SAT Prep, College geometry

Let A, B, C, D, E, F be points on a circle and makes a hexagon out of them in an arbitrary order. Then the three points L, M, N at which pairs of opposite sides meet, lie on a straight line. This line is called the Pascal line of the hexagon ABCDEF.

 

 


Proof:

1. Applying Menelaus’ Theorem respectively to transversal BCM, AFN, and DEL of triangle XYZ:


2. Multiplying the three identities, we get:


3. Using the intersecting secant theorem, we get:


4. By substituting (3) into (2), we get:



Thus, by the converse of Menelaus' Theorem, points M, N, and L lie on a transversal of triangle XYZ, therefore must be collinear. Q.E.D.
 


Blaise Pascal
(1623-1662) French mathematician, philosopher and inventor, discovered his famous theorem at the age of 16, in 1640, and produced a treatise on conic sections entitled Essai pour les coniques.

 

 

See also:

 

1.

Menelaus' Theorem. Interactive proof with animation and key concepts.

2.

Ceva's theorem

Ceva's Theorem. Concurrency. Interactive proof with animation. Key concept: Menelaus Theorem.

Puzzle of the Ceva's Theorem: 48 classic piece.

4.

Blanchet Theorem

5.

Nagel Point Theorem. Proof.
 

6.

Gergonne Point Theorem

Gergonne Point Theorem. Concurrency. Interactive proof with animation.

Key concept: Ceva's Theorem.
 

7.

Triangle with the bisectors of the exterior angles

Triangle with the bisectors of the exterior angles. Collinearity.Key concept: Menelaus Theorem.
 

8.

Monge & d'Alembert Three Circles Theorem I with Dynamic Geometry

9.

Monge & d'Alembert Three Circles Theorem II with Dynamic Geometry

10.

Newton's Theorem: Newton's Line. Circumscribed quadrilateral, midpoints of diagonals, center of the circle inscribed.

Puzzle of the Newton's Theorem: 50 pieces of circles.

11.

Pentagons & Pentagrams. I found, exploring with dynamic geometry software, new facts about pentagons and pentagrams. Key concept: Menelaus Theorem.

See also: Pentagons & Pentagrams - Puzzle: 96 triangles.

12.

Eyeball Theorem: Animated Angle to Geometry Study.

13.

Seven Circles Theorem

14.

Equal Incircles Theorem

15.

Archimedes' Book of Lemmas
 

16.

Butterfly Theorem

17.

Triangle Centers

18.

Euler and his beautiful and extraordinary formula

19.

Semiperimeter and incircle

20.

Semiperimeter and excircles of a triangle

21.

Semiperimeter, incircle and excircles of a triangle

22.

Miquel's Pentagram with Dynamic Geometry

23.

Miquel's Pentagram Theorem

Miquel's Pentagram Theorem. Proof

24.

Johnson's Theorem

25.

Simson Line

26.

Angle between two Simson Lines

27.

Sangaku

Sangaku Problem

28.

The Bevan Point

29.

Clifford's Circle Chain Theorems

30.

Kurschak's Tile and Theorem

31.

Animated Angle to Geometry Study

32.

Proposed Problem 28
Right Triangle, altitude, incircles and inradius.

33.

Steiner Point
 

 


 

 
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Last updated: February 13, 2009