Today we will learn about a very beautiful theorem in geometry -- the Star of David theorem. This theorem is a consequence of the Pascal hexagon theorem and the Pappus theorem.
Star of David theorem. Given two triangles $abc$ and $xyz$ inscribed in a same circle and six intersection points: $$p = ab \cap yz, ~~s = bc \cap zx, ~~q = ca \cap yz, ~~r = bc \cap xy, ~~1 = ab \cap xy, ~~3 = ca \cap zx.$$ Then the three lines $ps$, $qr$ and $13$ must meet at a common point.
To show that the three lines $ps$, $qr$, $13$ meet at a common point, we let $ps \cap qr = 2$, and prove that the three points $1$, $2$ and $3$ lie on the same line.
Observing the above figure, we can see that there are a lot of features look very similar to the Pascal hexagon theorem and the Pappus theorem.
Thus, we will use the Pascal hexagon theorem and the Pappus theorem to prove the Star of David theorem.
Draw the following intersection points: $$4 = yc \cap bz, ~~ 5 = pc \cap rz, ~~ 6 = ys \cap bq.$$
We will prove that the six points $1$, $2$, $3$, $4$, $5$ and $6$ are collinear.
Here is a sketch of a proof:
- The three points $1$, $4$, $3$ are collinear by the Pascal hexagon theorem
- The three points $1$, $4$, $5$ are collinear by the Pappus theorem
- The three points $3$, $5$, $2$ are collinear by the Pappus theorem
- The three points $1$, $2$, $6$ are collinear by the Pappus theorem
- The three points $3$, $4$, $6$ are collinear by the Pappus theorem
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| $1$, $4$, $3$ are collinear by the Pascal hexagon theorem |
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| $1$, $4$, $5$ are collinear by the Pappus theorem |
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| $3$, $5$, $2$ are collinear by the Pappus theorem |
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| $1$, $2$, $6$ are collinear by the Pappus theorem |
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| $3$, $4$, $6$ are collinear by the Pappus theorem |
The Star of David theorem for the conics
We know that the Pascal hexagon theorem holds for all types of conics and the Star of David theorem is a consequence of the Pascal hexagon theorem, therefore, the Star of David theorem also holds for conics. It means that, instead of choosing two triangles on a circle, we can choose two triangles on an ellipse, or on a parabola, or on a hyperbola, and the theorem is still true!
Here is a picture of an ellipse:
Here is a parabola:
And here is a hyperbola:
Let us stop here for now. Hope to see you again in our next post.
Homework.
1. Draw the Star of David theorem for different configurations by choosing various positions for the two triangles.
2. Write a full proof for the Star of David theorem based on our proof sketch above.
3. Go to google.com and search for other applications of the Pascal hexagon theorem.
