Home > Problems > Circles >

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. 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. Concurrency. Interactive proof with animation. Key concept: Ceva's Theorem.
7.		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. Proof
24.		Johnson's Theorem
25.		Simson Line
26.		Angle between two Simson Lines
27.		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